Measure Theory: Volume 1: The Irreducible Minimum

Measure Theory 1_The Irreducible Minimum(2e,2011,102p)D.H.Fremlin_9780953812981
	Contents
		General Introduction
		Introduction to Volume 1
		Note on second and third printings
		Note on second edition, 2011
	Chapter 11. Measure spaces
		111. σ-algebras
			111A. Definition
			111B. Remarks
			111C. Infinite unions and intersections
			111D. Elementary properties of σ-algebras
			111E. More on infinite unions and intersections
			111F. Countable sets
			111G. Borel sets
			111X. Basic exercises
			111Y. Further exercises
			111 Notes and comments
		112. Measure spaces
			112A. Definition
			112B. Remarks
			112C. Elementary properties of measure spaces
			112D. Negligible sets
			112X. Basic exercises
			112Y. Further exercises
			112 Notes and comments
		113. Outer measures and Carath éodory's construction
			113A. Outer measures
			113B. Remarks
			113C. Carathéodory's Method
			113D. Remark
			113X. Basic exercises
			113Y. Further exercises
			113 Notes and comments
		114. Lebesgue measure on R
			114A. Definitions
			114B. Lemma
			114C. Definition
			114D. Proposition
			114E. Definition
			114F. Lemma
			114G. Proposition
			114X. Basic exercises
			114Y. Further exercises
			114 Notes and comments
		115. Lebesgue measure on R^r
			115A. Definitions
			115B. Lemma
			115C. Definition
			115D. Proposition
			115E. Definition
			115F. Lemma
			115G. Proposition
			115X. Basic exercises
			115Y. Further exercises
			115 Notes and comments
	Chapter 12. Integration
		121. Measurable functions
			121A. Lemma
			121B. Proposition
			121C. Definition
			121D. Proposition
			121E. Theorem
			121F. Theorem
			121G. Remarks
			121H. Proposition
			*121I.
			*121J.
			*121K. Proposition
			121X. Basic exercises
			121Y. Further exercises
			121 Notes and comments
		122. Definition of the integral
			122A. Definitions
			122B. Lemma
			122C. Lemma
			122D. Corollary
			122E. Definition
			122F. Proposition
			122G. Lemma
			122H. Definition
			122I. Lemma
			122J. Lemma
			122K. Definition
			122L. Lemma
			122M. Definition
			122N. Remarks
			122O. Theorem
			122P. Theorem
			122Q. Remark
			122R. Corollary
			122X. Basic exercises
			122Y. Further exercises
			122 Notes and comments
		123. The convergence theorems
			123A. B.Levi's theorem
			123B. Fatou's Lemma
			123C. Lebesgue's Dominated Convergence Theorem
			123D. differentiating through an integral
			123X. Basic exercises
			123Y. Further exercises
			123 Notes and comments
	Chapter 13. Complements
		131. Measurable subspaces
			131A. Proposition
			131B. Definition
			131C. Lemma
			131D. Integration over subsets: Definition
			131E. Proposition
			131F. Corollary
			131G. Corollary
			131H. Corollary
			131X. Basic exercises
			131Y. Further exercises
			131 Notes and comments
		132. Outer measures from measures
			132A. Proposition
			132B. Definition
			132C. Proposition
			132D. Measurable envelopes
			132E. Lemma
			132F. Full outer measure
			132X. Basic exercises
			132Y. Further exercises
			132 Notes and comments
		133. Wider concepts of integration
			133A. Infinite integrals
			133B. Functions with exceptional values
			133C. Complex-valued functions
			133D. Definitions
			133E. Lemma
			133F. Proposition
			133G. Lebesgue's Dominated Convergence Theorem
			133H. Corollary
			133I. Upper and lower integrals
			133J. Proposition
			133K. Convergence theorems for upper integrals
			*133L.
			133X. Basic exercises
			133Y. Further exercises
			133 Notes and comments
		134. More on Lebesgue measure
			134A. Proposition
			134B. Theorem
			*134C. Remark
			*134D.
			134E. Borel sets and Lebesgue measure on R^r
			134F. Proposition
			134G. The Cantor set
			134H. The Cantor function
			134I. The Cantor function modified
			134J. More examples
			*134K. Riemann integration
			*134L.
			134X. Basic exercises
			134Y. Further exercises
			134 Notes and comments
		135. The extended real line
			135A. The algebraic structure of [-∞, ∞]
			135B. The order structure of [-∞,. ∞]
			135C. The Borel structure of [-∞,. ∞]
			135D. Convergent sequences in [-∞,. ∞]
			135E. Measurable functions
			135F.  [-∞,. ∞]-valued integrable functions
			135G.
			135H. Upper and lower integrals again
			135I. Subspace measures
			135X. Basic exercises
			135Y. Further exercises
			135 Notes and comments
		*136. The Monotone Class Theorem
			136A. Lemma
			136B. Monotone Class Theorem
			136C. Corollary
			136D. Corollary
			136E. Algebras of sets: Definition
			136F. Remarks
			136G. Theorem
			*136H. Proposition
			136X. Basic exercises
			136Y. Further exercises
			136 Notes and comments
	Appendix to Volume 1 - Useful Facts
		1A1. Set theory
			1A1A. Square bracket notations
			1A1B. Direct and inverse images
			1A1C. Countable sets
			1A1D. Proposition
			1A1E. Properties of countable sets
			1A1F.
			*1A1G. Remark
			1A1H. Some uncountable sets
			1A1I. Remark
			1A1J. Notation
			1A1 Notes and comments
		1A2. Open and closed sets in R^r
			1A2A. Open sets
			1A2B. The family of all open sets
			1A2C. Cauchy's inequality
			1A2D. Corollary
			1A2E Closed sets
			1A2F. Proposition
			1A2G.
		1A3. Lim sups and lim infs
			1A3A. Definition
			1A3B.
			1A3C. Remark
			*1A3D. Other expressions of the same idea
	Concordance
	References for Volume 1
	Index to volume 1
		Principal topics and results
		General index
Measure Theory 2_Broad Foundations(2016,570p)D.H.Fremlin_9780953812974
	Contents
		General introduction
		Introduction to Volume 2
		Note on second printing, April 2003
		Note on hardback edition, January 2010
		Note on second printing of hardback edition, April 2016
	*Chapter 21. Taxonomy of measure spaces
		211. Definitions
			211A. Definition
			211B. Definition
			211C. Definition
			211D. Definition
			211E. Definition
			211F. Definition
			211G. Definition
			211H. Definition
			211I. Definition
			211J. Definition
			211K. Definition
			211L.
			211M. Example: Lebesgue measure
			211N. Counting measure
			211O. A non-semi-finite space
			211P. A non-complete space
			211Q. Some probability spaces
			211R. Countable-cocountable measure
			211X. Basic exercises
			211Y. Further exercises
			211 Notes and comments
		212. Complete spaces
			212A. Proposition
			212B. Proposition
			212C. The completion of a measure
			212D.
			212E.
			212F.
			212G.
			212X. Basic exercises
			212Y. Further exercises
			212 Notes and comments
		213. Semi-finite, locally determined and localizable spaces
			213A. Lemma
			213B. Proposition
			*213C. Proposition
			213D. C.l.d. versions
			213E. Definition
			213F.
			213G.
			213H.
			213I. Locally determined negligible sets
			213J. Proposition
			*213K. Lemma
			213L. Proposition
			213M. Corollary
			213N.
			213O.
			213X. Basic exercises
			213Y. Further exercises
			213 Notes and comments
		214. Subspaces
			214A. Proposition
			214B. Definition
			214C. Lemma
			214D. Integration over subsets
			214E. Proposition
			214F. Proposition
			214G. Corollary
			214H. Subspaces and Carathéodory's method
			214I.
			214J. Upper and lower integrals
			214K. Measurable subspaces
			214L. Direct sums
			214M. Proposition
			214N. Corollary
			*214O.
			*214P. Theorem
			*214Q. Proposition
			214X. Basic exercises
			214Y. Further exercises
			214 Notes and comments
		215 σ-finite spaces and the principle of exhaustion
			215A. The principle of exhaustion
			215B.
			215C. Corollary
			215D.
			*215E.
			215X. Basic exercises
			215Y. Further exercises
			215 Notes and comments
		216. Examples
			216A. Lebesgue measure
			216B.
			*216C. A complete, localizable, non-locally-determined space
			*216D. A complete, locally determined space which is not localizable
			*216E. A complete, locally determined, localizable space which is not strictly localizable
			216X. Basic exercises
			216Y. Further exercises
			216 Notes and comments
	Chapter 22. The Fundamental Theorem of Calculus
		221. Vitali's theorem in R
			221A. Vitali's theorem
			221B. Remarks
			221X. Basic exercises
			221Y. Further exercises
			221 Notes and comments
		222. Differentiating an indefinite integral
			222A. Theorem
			222B. Remarks
			222C. Lemma
			222D. Lemma
			222E. Theorem
			222F. Corollary
			222G. Corollary
			222H.
			222I. Complex-valued functions
			*222J. The Denjoy-Young-Saks theorem
			*222K. Lemma
			*222L. Theorem
			222X. Basic exercises
			222Y. Further exercises
			222 Notes and comments
		223. Lebesgue's density theorems
			223A. Lebesgue's Density Theorem
			223B. Corollary
			223C. Corollary
			223D. Theorem
			223E. Complex-valued functions
			223X. Basic exercises
			223Y. Further exercises
			223 Notes and comments
		224. Functions of bounded variation
			224A. Definition
			224B. Remarks
			224C. Proposition
			224D. Theorem
			224E. Corollary
			224F. Corollary
			224G. Corollary
			224H. Proposition
			224I. Theorem
			224J.
			224K. Complex-valued functions
			224X. Basic exercises
			224Y. Further exercises
			224 Notes and comments
		225. Absolutely continuous functions
			225A. Absolute continuity of the indefinite integral
			225B. Absolutely continuous functions on R
			225C. Proposition
			225D. Lemma
			225E. Theorem
			225F. Integration by parts
			225G.
			225H. Semi-continuous functions
			225I. Proposition
			225J.
			225K. Proposition
			225L. Corollary
			225M. Corollary
			225N. The Cantor function
			225O. Complex-valued functions
			225X. Basic exercises
			225Y. Further exercises
			225 Notes and comments
		226. The Lebesgue decomposition of a function of bounded variation
			226A. Sums over arbitrary index sets
			226B. Saltus functions
			226C. The Lebesgue decomposition of a function of bounded variation
			226D. Complex functions
			226E.
			226X. Basic exercises
			226Y. Further exercises
			226 Notes and comments
	Chapter 23. The Radon-Nikodym Theorem
		231. Countably additive functionals
			231A. Definition
			231B. Elementary facts
			231C. Definition
			231D. Elementary facts
			231E.
			231F. Corollary
			231X. Basic exercises
			231Y. Further exercises
			231 Notes and comments
		232. The Radon-Nikodým theorem
			232A. Absolutely continuous functionals
			232B. Proposition
			232C. Lemma
			232D. Proposition
			232E. The Radon-Nikodým theorem
			232F. Corollary
			232G. Corollary
			232H. Remarks
			232I. The Lebesgue decomposition of a countably additive functional
			232X. Basic exercises
			232Y. Further exercises
			232 Notes and comments
		233. Conditional expectations
			233A. σ-subalgebras
			233B. Lemma
			233C. Remarks
			233D. Conditional expectations
			233E.
			233F. Remarks
			233G. Convex functions
			233H.
			233I. Jensen's inequality
			233J.
			233K.
			233X. Basic exercises
			233Y. Further exercises
			233 Notes and comments
		234. Operations on measures
			234A. Inverse-measure-preserving functions
			234B. Proposition
			234C. Image measures
			234D. Definition
			234E. Proposition
			*234F.
			234G. Sums of measures
			234H. Proposition
			234I. Indefinite-integral measures
			234J. Definition
			234K. Remarks
			234L. The domain of an indefinite-integral measure
			234M. Corollary
			*234N.
			*234O.
			234P. Ordering measures
			234Q. Proposition
			234X. Basic exercises
			234Y. Further exercises
			234 Notes and comments
		235. Measurable transformations
			235A.
			235B. Remarks
			235C.
			235D.
			235E.
			235F. Remarks
			235G. Theorem
			235H. The image measure catastrophe
			235I. Lemma
			235J. Theorem
			235K. Corollary
			235L. Applying the Radon-Nikodým theorem
			235M. Theorem
			235N. Remark
			*235O.
			*235P. Proposition
			*235Q.
			235R. Reversing the burden
			235X. Basic exercises
			235Y. Further exercises
			235 Notes and comments
	Chapter 24. Function spaces
		241. \usepackage{euscript} \mathscr{L}^0 and L^0
			241A. The space \usepackage{euscript} \mathscr{L}^0 : Definition
			241B. Basic properties
			241C. The space L^0: Definition
			241D. The linear structure of L^0
			241E. The order structure of L^0
			241F. Riesz spaces
			241G.
			241H. The multiplicative structure of L^0
			241I. The action of Borel functions on L^0
			241J. Complex L^0
			241X. Basic exercises
			241Y. Further exercises
			241 Notes and comments
		242. L^1
			242A. The space L^1
			242B. Theorem
			242C. The order structure of L^1
			242D. The norm of L^1
			242E.
			242F. Theorem
			242G. Definition
			242H. L^1 as a Riesz space
			242I. The Radon-Nikodým theorem
			242J. Conditional expectations revisited
			242K.
			242L. Proposition
			242M. L^1 as a completion
			242N.
			242O. Theorem
			242P. Complex L^1
			242X. Basic exercises
			242Y. Further exercises
			242 Notes and comments
		243. L^∞
			243A. Definitions
			243B. Theorem
			243C. The order structure of L^∞
			243D The norm of L^∞
			243E. Theorem
			243F. The duality between L^∞ and L^1
			243G. Theorem
			243H.
			243I. A dense subspace of  L^∞
			243J. Conditional expectations
			243K. Complex L^∞
			243X. Basic exercises
			243Y. Further exercises
			243 Notes and comments
		244. L^p
			244A. Definitions
			244B. Theorem
			244C. The order structure of L^p
			244D. The norm of L^p
			244E.
			244F. Proposition
			244G. Theorem
			244H.
			*244I. Corollary
			244J. Duality in L^p
			244K. Theorem
			244L.
			244M.
			244N. The space L^2
			*244O.
			244P. Complex L^p
			244X. Basic exercises
			244Y. Further exercises
			244 Notes and comments
		245. Convergence in measure
			245A. Definitions
			245B. Remarks
			245C. Pointwise convergence
			245D. Proposition
			245E.
			245F. Alternative description of the topology of convergence in measure
			245G. Embedding L^p in L^0
			245H.
			245I. Remarks
			245J.
			245K.
			245L. Corollary
			245M. Complex L^0
			245X. Basic exercises
			245Y. Further exercises
			245 Notes and comments
		246. Uniform integrability
			246A. Definition
			246B. Remarks
			246C.
			246D. Proposition
			246E. Remarks
			246F.
			246G.
			246H. Remarks
			246I. Corollary
			246J.
			246K. Complex \usepackage{euscript} \mathscr{L}^1 and L^1
			246X. Basic exercises
			246Y. Further exercises
			246 Notes and comments
		247. Weak compactness in L^1
			247A.
			247B. Corollary
			247C. Theorem
			247D. Corollary
			247E. Complex L^1
			247X. Basic exercises
			247Y. Further exercises
			247 Notes and comments
	Chapter 25. Product Measures
		251. Finite products
			251A. Definition
			251B. Lemma
			251C. Definition
			251D. Definition
			251E. Proposition
			251F. Definition
			251G. Remark
			251H.
			251I.
			251J. Proposition
			251K. σ-finite spaces
			*251L.
			251M.
			251N. Theorem
			251O.
			251P. Lemma
			251Q. Proposition
			251R. Corollary
			251S. Corollary
			251T.
			251U.
			*251W. Products of more than two spaces
			251X. Basic exercises
			251Y. Further exercises
			251 Notes and comments
		252. Fubini's theorem
			252A. Repeated integrals
			252B. Theorem
			252C.
			252D.
			252E. Corollary
			252F.
			252G.
			252H. Corollary
			252I. Corollary
			252J. Remarks
			252K. Example
			252L. Example
			252M. Remark
			252N. Integration through ordinate sets I
			252O. Integration through ordinate sets II
			*252P.
			252Q. The volume of a ball
			252R. Complex-valued functions
			252X. Basic exercises
			252Y. Further exercises
			252 Notes and comments
		253. Tensor products
			253A. Bilinear operators
			253B. Definition
			253C. Proposition
			253D.
			253E. The canonical map L^1 × L^1 -> L^1
			253F.
			253G. The order structure of L^1
			253H. Conditional expectations
			253I.
			*253J. Upper integrals
			*253K.
			253L. Complex spaces
			253X. Basic exercises
			253Y. Further exercises
			253 Notes and comments
		254. Infinite products
			254A. Definitions
			254B. Lemma
			254C. Definition
			254D. Remarks
			254E. Definition
			254F. Theorem
			254G.
			254H. Corollary
			254I.
			254J. The product measure on {0,1}^I
			254K.
			254L. Subspaces
			254M.
			254N. Theorem
			254O. Proposition
			254P. Proposition
			254Q. Proposition
			254R. Conditional expectations again
			254S. Proposition
			254T. Remarks
			*254U.
			*254V.
			254X. Basic exercises
			254Y. Further exercises
			254 Notes and comments
		255. Convolutions of functions
			255A.
			255B. Corollary
			255C. Remarks
			255D.
			255E. The basic formula
			255F. Elementary properties
			255G.
			255H.
			255I. Corollary
			255J. Theorem
			255K.
			255L. The r-dimensional case
			255M. The case of ]-π, π]
			255N. Theorem
			255O. Convolutions on ]-π, π]
			255X. Basic exercises
			255Y. Further exercises
			255 Notes and comments
		256. Radon measures on R^r
			256A. Definitions
			256B.
			256C. Theorem
			256D. Proposition
			256E.
			256F. Theorem
			256G. Theorem
			256H. Examples
			256I. Remarks
			256J. Absolutely continuous Radon measures
			256K. Products
			256L. Remark
			*256M.
			256X. Basic exercises
			256Y. Further exercises
			256 Notes and comments
		257. Convolutions of measures
			257A. Definition
			257B. Theorem
			257C. Corollary
			257D. Corollary
			257E. Corollary
			257F. Theorem
			257X. Basic exercises
			257Y. Further exercises
			257 Notes and comments
	Chapter 26. Change of Variable in the Integral
		261. Vitali's theorem in R^r
			261A. Notation
			261B. Vitali's theorem in R^r
			261C.
			261D. Corollary
			261E. Theorem
			261F.
			261X. Basic exercises
			261Y. Further exercises
			261 Notes and comments
		262 Lipschitz and differentiable functions
			262A. Lipschitz functions
			262B.
			262C. Remark
			262D. Proposition
			262E. Corollary
			262F. Differentiability
			262G. Remarks
			262H. The norm of a matrix
			262I. Lemma
			262J. Remark
			262K. The Cantor function revisited
			262L.
			262M.
			262N. Corollary
			262O. Corollary
			262P. Corollary
			*262Q.
			262X. Basic exercises
			262Y. Further exercises
			262 Notes and comments
		263. Differentiable transformations in R^r
			263A. Linear transformations
			263B. Remark
			263C. Lemma
			263D.
			263E. Remarks
			*263F. Corollary
			263G. Polar coordinates in the plane
			263H. Corollary
			263I.
			263J. The one-dimensional case
			263X. Basic exercises
			263Y. Further exercises
			263 Notes and comments
		264 Hausdorff measures
			264A. Definitions
			264B.
			264C. Definition
			264D. Remarks
			264E. Theorem
			264F. Proposition
			264G. Lipschitz functions
			264H.
			264I. Theorem
			*264J. The Cantor set
			264X. Basic exercises
			264Y. Further exercises
			264 Notes and comments
		265. Surface measures
			265A. Normalized Hausdorff measure
			265B. Linear subspaces
			265C. Corollary
			265D.
			265E. Theorem
			265F. The surface of a sphere
			265G.
			265H. Corollary
			265X. Basic exercises
			265Y. Further exercises
			265 Notes and comments
		*266 .The Brunn-Minkowski inequality
			266A. Arithmetic and geometric means
			266B. Proposition
			266C. Theorem
			266X. Basic exercises
			266 Notes and comments
	Chapter 27. Probability theory
		271. Distributions
			271A. Notation
			271B. Theorem
			271C. Definition
			271D. Remarks
			271E. Measurable functions of random variables
			271F. Corollary
			271G. Distribution functions
			271H. Densities
			271I. Proposition
			271J.
			271K.
			*271L.
			271X. Basic exercises
			271Y. Further exercises
			271 Notes and comments
		272. Independence
			272A. Definitions
			272B. Remarks
			272C. The σ-subalgebra defined by a random variable
			272D. Proposition
			272E. Corollary
			272F.
			272G. Distributions of independent random variables
			272H. Corollary
			272I. Corollary
			272J.
			272K. Proposition
			272L.
			272M. Products of probability spaces and independent families of random variables
			272N.
			272O. Tail σ-algebras and the zero-one law
			272P.
			*272Q.
			272R.
			272S. Bienaymé's Equality
			272T. The distribution of a sum of independent random variables
			272U. Corollary
			272V.
			*272W.
			272X. Basic exercises
			272Y. Further exercises
			272 Notes and comments
		273. The strong law of large numbers
			273A.
			273B. Lemma
			273C.
			273D. The strong law of large numbers: first form
			273E. Corollary
			273F. Corollary
			273G. Corollary
			273H. Strong law of large numbers: second form
			273I. Strong law of large numbers: third form
			273J. Corollary
			273K.
			273L.
			*273M.
			273N. Theorem
			273X. Basic exercises
			273Y. Further exercises
			273 Notes and comments
		274. The central limit theorem
			274A. The normal distribution
			274B. Proposition
			274C. Lemma
			274D. Lemma
			274E. Lemma
			274F. Lindeberg's theorem
			274G. Central Limit Theorem
			274H. Remarks
			274I. Corollary
			274J. Corollary
			274K. Corollary
			274L. Remarks
			*274M.
			274X. Basic exercises
			274Y. Further exercises
			274 Notes and comments
		275. Martingales
			275A. Definition
			275B. Examples
			275C. Remarks
			275D.
			275E. Up-crossings
			275F. Lemma
			275G.
			275H. Theorem
			275I. Theorem
			*275J.
			275K. Reverse martingales
			275L. Stopping times
			275M. Examples
			275N. Lemma
			275O. Proposition
			275P. Corollary
			275X. Basic exercises
			275Y. Further exercises
			275 Notes and comments
		276. Martingale difference sequences
			276A. Martingale difference sequences
			276B. Proposition
			276C. The strong law of large numbers: fourth form
			276D. Corollary
			276E. `Impossibility of systems'
			*276F.
			*276G. Lemma
			*276H. Komlós' theorem (KOMLÓS 67)
			276X. Basic exercises
			276Y. Further exercises
			276 Notes and comments
	Chapter 28. Fourier analysis
		281. The Stone-Weierstrass theorem
			281A. Stone-Weierstrass theorem: first form
			281B.
			281C. Lemma
			281D. Corollary
			281E. Stone-Weierstrass theorem: second form
			281F. Corollary: Weierstrass' theorem
			281G. Stone-Weierstrass theorem: third form
			281H. Corollary
			281I. Corollary
			281J. Corollary
			281K. Corollary
			281L. Corollary
			281M. Weyl's Equidistribution Theorem
			281N. Theorem
			281X. Basic exercises
			281Y. Further exercises
			281 Notes and comments
		282. Fourier series
			282A. Definition
			282B. Remarks
			282C. The problems
			282D. Lemma
			282E.
			282F. Corollary
			282G.
			282H.
			282I. Corollary
			282J.
			282K. Corollary
			282L.
			282M. Lemma
			282N.
			282O. Theorem
			282P. Corollary
			282Q.
			*282R.
			282X. Basic exercises
			282Y. Further exercises
			282 Notes and comments
		283. Fourier transforms I
			283A. Definitions
			283B. Remarks
			283C. Proposition
			283D. Lemma
			283E.
			283F. Theorem
			283G. Corollary
			283H. Lemma
			283I. Theorem
			283J. Corollary
			283K.
			283L.
			283M.
			283N.
			283O.
			283W. Higher dimensions
			283X. Basic exercises
			283Y. Further exercises
			283 Notes and comments
		284. Fourier transforms II
			284A. Test functions: Definition
			284B.
			284C. Proposition
			284D. Definition
			284E.
			284F.
			284G. Lemma
			284H. Definition
			284I. Remarks
			284J. Lemma
			284K. Proposition
			284L.
			284M. Theorem
			284N. L^2 spaces
			284O. Theorem
			284P. Corollary
			284Q. Remarks
			284R. Dirac's delta function
			284W. The multidimensional case
			284X. Basic exercises
			284Y. Further exercises
			284 Notes and comments
		285. Characteristic functions
			285A. Definition
			285B. Remarks
			285C.
			285D.
			285E. Lemma
			285F.
			285G. Corollary
			285H. Remark
			285I. Proposition
			285J.
			285K. Characteristic functions and the vague topology
			285L. Theorem
			285M. Corollary
			285N. Remarks
			285O. Lemma
			285P. Lemma
			285Q. Law of Rare Events: Theorem
			285R. Convolutions
			285S. The vague topology and pointwise convergence of characteristic functions
			285T. Proposition
			285U. Corollary
			285X. Basic exercises
			285Y. Further exercises
			285 Notes and comments
		286. Carleson's theorem
			286A. The Maximal Theorem
			286B. Lemma
			286C. Shift, modulation and dilation
			286D. Lemma
			286E. The Lacey-Thiele construction
			286F. A partial order
			286G.
			286H. `Mass' and `energy' (Lacey & Thiele 00)
			286I. Lemma
			286J. Lemma
			286K. Lemma
			286L. Lemma
			286M. The Lacey-Thiele lemma
			286N. Lemma
			286O. Lemma
			286P. Lemma
			286Q. Lemma
			286R. Lemma
			286S. Lemma
			286T. Lemma
			286U. Theorem
			286V. Theorem
			286W. Glossary
			286X. Basic exercises
			286Y. Further exercises
			286 Notes and comments
	Appendix to Volume 2 - Useful Facts
		2A1. Set theory
			2A1A. Ordered sets
			2A1B. Transfinite Recursion: Theorem
			2A1C. Ordinals
			2A1D. Basic facts about ordinals
			2A1E. Initial ordinals An initial ordinal
			2A1F. Basic facts about initial ordinals
			2A1G. Schröder-Bernstein theorem
			2A1H. Countable subsets of PN
			2A1I. Filters
			2A1J. The Axiom of Choice
			2A1K. Zermelo's Well-Ordering Theorem
			2A1L. Fundamental consequences of the Axiom of Choice
			2A1M. Zorn's Lemma
			2A1N. Ultrafilters
			2A1O. The Ultrafilter Theorem
			2A1P.
		2A2. The topology of Euclidean space
			2A2A. Closures: Definition
			2A2B. Lemma
			2A2C. Continuous functions
			2A2D. Compactness in R^r: Definition
			2A2E. Elementary properties of compact sets
			2A2F.
			2A2G. Corollary
			2A2H. Lim sup and lim inf revisited
			2A2I.
		2A3. General topology
			2A3A. Topologies
			2A3B. Continuous functions
			2A3C. Subspace topologies
			2A3D. Closures and interiors
			2A3E Hausdorff topologies
			2A3F. Pseudometrics
			2A3G. Proposition
			2A3H.
			2A3I. Remarks
			2A3J. Subspaces: Proposition
			2A3K. Closures and interiors
			2A3L. Hausdorff topologies
			2A3M. Convergence of sequences
			2A3N. Compactness
			2A3O. Cluster points
			2A3P. Filters
			2A3Q. Convergent filters
			2A3R.
			2A3S. Further calculations with filters
			2A3T. Product topologies
			2A3U. Dense sets
		2A4. Normed spaces
			2A4A. The real and complex fields
			2A4B. Definitions
			2A4C. Linear subspaces
			2A4D. Banach spaces
			2A4E.
			2A4F. Bounded linear operators
			2A4G. Theorem
			2A4H. Dual spaces
			2A4I. Extensions of bounded operators: Theorem
			2A4J. Normed algebras
			*2A4K. Definition
		2A5. Linear topological spaces
			2A5A. Linear space topologies
			2A5B.
			*2A5C.
			2A5D. Definition
			2A5E. Convex sets
			2A5F. Completeness in linear topological spaces
			2A5G.
			2A5H. Normed spaces and sequential completeness
			2A5I. Weak topologies
			*2A5J. Angelic spaces
		2A6. Factorization of matrices
			2A6A. Determinants
			2A6B. Orthonormal families
			2A6C.
	Concordance
	References for Volume 2
	Index to volumes 1 and 2
		Principal topics and results
		General index
Measure Theory 3-1_Measure Algebras(2e,2012,216p)D.H.Fremlin_9780956607102
	Contents
		General introduction
		Introduction to Volume 3
		Note on second printing
		Note on second (`Lulu') edition
	Chapter 31. Boolean algebras
		311. Boolean algebras
			311A. Definitions
			311B. Examples
			311C. Proposition
			311D. Lemma
			311E. M.H.Stone's theorem: first form
			311F. Remarks
			311G. The operations ∪, ＼, Δ on a Boolean ring
			311H. The order structure of a Boolean ring
			311I. The topology of a Stone space: Theorem
			311J.
			311K. Remark
			311L. Complemented distributive lattices
			311X. Basic exercises
			311Y. Further exercises
			311 Notes and comments
		312. Homomorphisms
			312A. Subalgebras
			312B. Proposition
			312C. Ideals in Boolean algebras: Proposition
			312D. Principal ideals
			312E. Proposition
			312F. Boolean homomorphisms
			312G. Proposition
			312H. Proposition
			312I. Proposition
			312J. Proposition
			*312K. Fixed-point subalgebras
			312L. Quotient algebras: Proposition
			312M.
			312N.
			312O. Lemma
			312P. Homomorphisms and Stone spaces
			312Q. Theorem
			312R. Theorem
			312S. Proposition
			312T. Principal ideals
			312X. Basic exercises
			312Y. Further exercises
			312 Notes and comments
		313. Order-continuous homomorphisms
			313A. Relative complementation: Proposition
			313B. General distributive laws: Proposition
			313C.
			313D.
			313E. Order-closed subalgebras and ideals
			313F. Order-closures and generated sets
			313G.
			313H. Definitions
			313I. Proposition
			313J.
			313K. Lemma
			313L. Proposition
			313M.
			313N. Definition
			313O. Proposition
			313P.
			313Q. Corollary
			313R.
			313S. Upper envelopes
			313X. Basic exercises
			313Y. Further exercises
			313 Notes and comments
		314. Order-completeness
			314A. Definitions
			314B. Remarks
			314C. Proposition
			314D. Corollary
			314E. Proposition
			314F.
			314G. Corollary
			314H. Corollary
			314I. Corollary
			314J.
			314K. Extension of homomorphisms
			314L. The Loomis-Sikorski representation of a Dedekind σ-complete Boolean algebra
			314M. Theorem
			314N. Corollary
			314O. Regular open algebras
			314P. Theorem
			314Q. Remarks
			*314R.
			314S.
			314T.
			314U. The Dedekind completion of a Boolean algebra
			314X. Basic exercises
			314Y. Further exercises
			314 Notes and comments
		315. Products and free products
			315A. Products of Boolean algebras
			315B. Theorem
			315C. Products of partially ordered sets
			315D. Proposition
			315E. Factor algebras as principal ideals
			315F. Proposition
			315G. Algebras of sets and their quotients
			*315H.
			315I. Free products
			315J. Theorem
			315K.
			315L. Proposition
			315M. Algebras of sets and their quotients
			315N. Notation
			315O. Lemma
			315P. Example
			315Q. Example
			*315R. Projective and inductive limits: Proposition
			*315S. Definitions
			315X. Basic exercises
			315Y. Further exercises
			315 Notes and comments
		316. Further topics
			316A. Definitions
			316B. Theorem
			316C. Proposition
			316D. Corollary
			316E. Proposition
			316F. Corollary
			316G. Definition
			316H. Proposition
			316I.
			316J. The regular open algebra of R
			316K. Atoms in Boolean algebras
			316L. Proposition
			316M. Proposition
			316N. Definition
			*316O. Lemma
			*316P. Proposition
			*316Q. Proposition
			316X. Basic exercises
			316Y. Further exercises
			316 Notes and comments
	Chapter 32. Measure algebras
		321. Measure algebras
			321A. Definition
			321B. Elementary properties of measure algebras
			321C. Proposition
			321D. Corollary
			321E. Corollary
			321F. Corollary
			321G. Subalgebras
			321H. The measure algebra of a measure space
			321I. Definition
			321J. The Stone representation of a measure algebra
			321K. Definition
			321X. Basic exercises
			321Y. Further exercises
			321 Notes and comments
		322. Taxonomy of measure algebras
			322A. Definitions
			322B.
			322C.
			322D.
			322E. Proposition
			322F. Proposition
			322G.
			322H. Principal ideals
			322I. Subspace measures
			322J. Corollary
			322K. Indefinite-integral measures
			322L. Simple products
			*322M. Strictly localizable spaces
			322N. Subalgebras
			322O. The Stone space of a localizable measure algebra
			322P. Theorem
			322Q. Definition
			322R. Further properties of Stone spaces
			322X. Basic exercises
			322Y. Further exercises
			322 Notes and comments
		323. The topology of a measure algebra
			323A. The pseudometrics
			323B. Proposition
			323C. Proposition
			323D.
			323E. Corollary
			323F.
			323G. The classification of measure algebras
			323H. Closed subalgebras
			323I. Notation
			323J. Proposition
			323K.
			323L. Proposition
			*323M.
			323X. Basic exercises
			323Y. Further exercises
			323 Notes and comments
		324. Homomorphisms
			324A. Theorem
			324B. Corollary
			324C. Remarks
			324D. Proposition
			324E. Stone spaces
			324F.
			324G. Corollary
			324H. Corollary
			324I. Definition
			324J. Proposition
			324K. Proposition
			324L. Corollary
			324M. Proposition
			324N. Proposition
			324O. Proposition
			*324P.
			324X. Basic exercises
			324Y. Further exercises
			324 Notes and comments
		325. Free products and product measures
			325A. Theorem
			325B. Characterizing the measure algebra of a product space
			325C.
			325D. Theorem
			325E. Remarks
			325F.
			325G.
			*325H. Products of more than two factors
			325I. Infinite products
			325J.
			325K. Definition
			325L. Independent subalgebras
			325M.
			*325N. Notation
			325X. Basic exercises
			325Y. Further exercises
			325 Notes and comments
		326. Additive functionals on Boolean algebras
			326A. Additive functionals
			326B. Elementary facts
			326C. The space of additive functionals
			326D. The Jordan decomposition (I)
			*326E. Additive functionals on free products
			*326F.
			*326G. Lemma
			*326H Liapounoff's convexity theorem (LIAPOUNOFF 40)
			326I. Countably additive functionals
			326J. Elementary facts
			326K. Corollary
			326L. The Jordan decomposition (II)
			326M. The Hahn decomposition
			326N. Completely additive functionals
			326O. Basic facts
			326P.
			326Q. The Jordan decomposition (III)
			326R.
			326S.
			326T. Corollary
			326X. Basic exercises
			326Y. Further exercises
			326 Notes and comments
		327. Additive functionals on measure algebras
			327A.
			327B. Theorem
			327C. Proposition
			327D. The Radon-Nikodým theorem
			327E.
			327F. Standard extensions
			327G. Definition
			327X. Basic exercises
			327Y. Further exercises
			327 Notes and comments
		*328 .Reduced products and other constructions
			328A. Construction
			328B. Proposition
			328C. Definition
			328D. Proposition
			328E. Proposition
			328F. Corollary
			328G. Corollary
			328H. Proposition
			328I.
			328J.
			328X. Basic exercises
			328 Notes and comments
	Chapter 33. Maharam's theorem
		331. Maharam types and homogeneous measure algebras
			331A. Definition
			331B.
			331C. Corollary
			331D. Lemma
			331E. Generating sets
			331F. Maharam types
			331G.
			331H. Proposition
			331I.
			331J. Lemma
			331K. Theorem
			331L. Theorem
			331M. Homogeneous Boolean algebras
			331N. Proposition
			331O.
			331X. Basic exercises
			331Y. Further exercises
			331 Notes and comments
		332. Classification of localizable measure algebras
			332A. Lemma
			332B. Maharam's theorem
			332C. Corollary
			332D. The cellularity of a Boolean algebra
			332E. Proposition
			332F. Corollary
			332G. Definitions
			332H. Lemma
			332I. Lemma
			332J.
			332K. Remarks
			332L. Proposition
			332M. Lemma
			332N. Lemma
			332O. Lemma
			332P. Proposition
			332Q. Proposition
			332R.
			332S. Theorem
			332T. Proposition
			332X. Basic exercises
			332Y. Further exercises
			332 Notes and comments
		333. Closed subalgebras
			333A. Definitions
			333B.
			333C. Theorem
			333D. Corollary
			333E. Theorem
			333F. Corollary
			333G. Corollary
			333H.
			333I. Remarks
			333J. Lemma
			333K. Theorem
			333L. Remark
			333M. Lemma
			333N. A canonical form for closed subalgebras
			333O. Remark
			333P.
			333Q. Corollary
			333R.
			333X. Basic exercises
			333Y. Further exercises
			333 Notes and comments
		334. Products
			334A. Theorem
			334B. Corollary
			334C. Theorem
			334D. Corollary
			334E.
			334X .Basic exercises
			334Y. Further exercises
			334 Notes and comments
	Chapter 34. The lifting theorem
		341. The lifting theorem
			341A. Definition
			341B. Remarks
			341C. Definition
			341D. Remarks
			341E. Example
			341F.
			341G. Lemma
			341H.
			341I.
			341J. Proposition
			341K. The Lifting Theorem
			341L. Remarks
			341M.
			341N. Extension of partial liftings
			341O. Liftings and Stone spaces
			341P. Proposition
			341Q. Corollary
			341X. Basic exercises
			341Y. Further exercises
			341Z. Problems
			341 Notes and comments
		342. Compact measure spaces
			342A. Definitions
			342B.
			342C. Corollary
			342D. Lemma
			342E. Corollary
			342F. Corollary
			342G.
			342H. Proposition
			342I. Proposition
			342J. Examples
			342K.
			342L. Theorem
			342M.
			*342N. Example
			342X. Basic exercises
			342Y. Further exercises
			342 Notes and comments
		343. Realization of homomorphisms
			343A. Preliminary remarks
			343B. Theorem
			343C. Examples
			343D. Uniqueness of realizations
			343E. Lemma
			343F. Proposition
			343G. Corollary
			343H. Examples
			343I. Example
			343J. The split interval
			343K.
			343L.
			343M. Example
			343X. Basic exercises
			343Y. Further exercises
			343 Notes and comments
		344. Realization of automorphisms
			344A. Stone spaces
			344B. Theorem
			344C. Corollary
			344D.
			344E. Theorem
			344F. Corollary
			344G. Corollary
			344H. Lemma
			344I. Theorem
			344J. Corollary
			344K. Corollary
			344L.
			344X. Basic exercises
			344Y. Further exercises
			344 Notes and comments
		345. Translation-invariant liftings
			345A. Translation-invariant liftings
			345B. Theorem
			345C. Theorem
			345D.
			345E.
			345F. Proposition
			345X. Basic exercises
			345Y. Further exercises
			345 Notes and comments
		346. Consistent liftings
			346A. Definition
			346B. Lemma
			346C. Theorem
			346D.
			346E. Theorem
			346F.
			346G. Theorem
			346H. Theorem
			346I. Theorem
			346J. Consistent liftings
			346K. Lemma
			346L. Proposition
			346X. Basic exercises
			346Y. Further exercises
			346Z. Problems
			346 Notes and comments
	Concordance
Measure Theory 3-2_Measure Algebras(2e,2012,469p)D.H.Fremlin_9780956607119
	Contents
	Chapter 35. Riesz spaces
		351. Partially ordered linear spaces
			351A. Definition
			351B. Elementary facts
			351C. Positive cones
			351D. Suprema and infima
			351E. Linear subspaces
			351F. Positive linear operators
			351G. Order-continuous positive linear operators
			351H. Riesz homomorphisms
			351I. Solid sets
			351J. Proposition
			351K. Lemma
			351L. Products
			351M. Reduced powers of R
			351N.
			351O. Lemma
			351P. Lemma
			351Q.
			351R. Archimedean spaces
			351X. Basic exercises
			351Y. Further exercises
			351 Notes and comments
		352. Riesz spaces
			352A.
			352B. Lemma
			352C. Notation
			352D. Elementary identities
			352E. Distributive laws
			352F. Further identities and inequalities
			352G. Riesz homomorphisms
			352H. Proposition
			352I. Riesz subspaces
			352J. Solid subsets
			352K. Products
			352L. Theorem
			352M. Corollary
			352N. Order-density and order-continuity
			352O. Bands
			352P. Complemented bands
			352Q. Theorem
			352R. Projection bands
			352S. Proposition
			352T. Products again
			352U. Quotient spaces
			352V. Principal bands
			352W. f-algebras
			352X. Basic exercises
			352 Notes and comments
		353. Archimedean and Dedekind complete Riesz spaces
			353A. Proposition
			353B. Proposition
			353C. Corollary
			353D. Proposition
			353E. Lemma
			353F. Lemma
			353G. Dedekind completeness
			353H. Proposition
			353I. Proposition
			353J. Proposition
			353K. Proposition
			353L. Order units
			353M. Theorem
			353N. Lemma
			353O. f-algebras
			353P. Proposition
			353Q. Proposition
			353X. Basic exercises
			353Y. Further exercises
			353 Notes and comments
		354. Banach lattices
			354A. Definitions
			354B. Lemma
			354C. Lemma
			354D.
			354E. Proposition
			354F. Lemma
			354G. Definitions
			354H. Examples
			354I. Lemma
			354J. Proposition
			354K. Theorem
			354L. Corollary
			354M.
			354N. Theorem
			354O. Proposition
			354P. Uniform integrability in L-spaces
			354Q.
			354R.
			354X. Basic exercises
			354Y. Further exercises
			354 Notes and comments
		355. Spaces of linear operators
			355A. Definition
			355B. Lemma
			355C. Theorem
			355D. Lemma
			355E. Theorem
			355F. Theorem
			355G. Definition
			355H. Theorem
			355I. Theorem
			355J. Proposition
			355K. Proposition
			355X. Basic exercises
			355Y. Further exercises
			355 Notes and comments
		356. Dual spaces
			356A. Definition
			356B. Theorem
			356C. Proposition
			356D. Proposition
			356E. Biduals
			356F. Theorem
			356G. Lemma
			356H. Lemma
			356I. Theorem
			356J. Definition
			356K. Proposition
			356L. Proposition
			356M. Proposition
			356N. L- and M-spaces
			356O. Theorem
			356P. Proposition
			356Q. Theorem
			356X. Basic exercises
			356Y. Further exercises
			356 Notes and comments
	Chapter 36. Function Spaces
		361. S
			361A. Boolean rings
			361B. Definition
			361C. Elementary facts
			361D. Construction
			361E.
			361F.
			361G. Theorem
			361H. Theorem
			361I. Theorem
			361J.
			361K. Proposition
			361L. Proposition
			361M. Proposition
			361X. Basic exercises
			361Y. Further exercises
			361 Notes and comments
		362. S~
			362A. Theorem
			362B. Spaces of finitely additive functionals
			362C.
			362D.
			362E. Uniformly integrable sets
			362X. Basic exercises
			362Y. Further exercises
			362 Notes and comments
		363. L^∞
			363A. Definition
			363B. Theorem
			363C. Proposition
			363D. Proposition
			363E. Theorem
			363F. Theorem
			363G. Corollary
			363H. Representations of L^\infty ( \mathfrak{A} )
			363I. Corollary
			363J. Recovering the algebra \mathfrak{A}
			363K. Dual spaces of L^\infty
			*363L. Integration with respect to a finitely additive functional
			363M.
			363N.
			363O. Corollary
			363P. Corollary
			363Q.
			363R.
			363S. The Banach-Ulam problem
			363X. Basic exercises
			363Y. Further exercises
			363 Notes and comments
		364. L^0
			364A. The set L^0( \mathfrak{A} )
			364B. Proposition
			364C. Theorem
			364D. Theorem
			364E.
			364F.
			364G. Definition
			364H. Proposition
			364I. Examples
			364J. Embedding S and L^\infty in L^0
			364K. Corollary
			364L. Suprema and infima in L^0
			364M.
			364N. The multiplication of L^0
			364O. Recovering the algebra
			364P.
			364Q. Proposition
			364R. Products
			*364S. Regular open algebras
			*364T. Theorem
			*364U. Compact spaces
			*364V. Theorem
			364X. Basic exercises
			364Y. Further exercises
			364 Notes and comments
		365. L^1
			365A. Definition
			365B. Theorem
			365C.
			365D. Integration
			365E. The Radon-Nikodým theorem again
			365F.
			365G. Semi-finite algebras
			365H. Measurable transformations
			365I. Theorem
			365J. Corollary
			365K. Theorem
			365L. The duality between L^1 and L^\infty
			365M. Theorem
			365N. Corollary
			365O. Theorem
			365P. Theorem
			365Q. Proposition
			365R. Conditional expectations
			365S. Recovering the algebra: Proposition
			365T.
			365U. Uniform integrability
			365X. Basic exercises
			365Y. Further exercises
			365 Notes and comments
		366. L^p
			366A. Definition
			366B. Theorem
			366C. Corollary
			366D.
			366E. Proposition
			366F.
			366G. Lemma
			366H. Theorem
			366I. Corollary
			366J. Corollary
			366K. Corollary
			366L. Corollary
			*366M. Complex L^p spaces
			366X. Basic exercises
			366Y. Further exercises
			366 Notes and comments
		367. Convergence in measure
			367A. Order*-convergence
			367B. Lemma
			367C. Proposition
			367D.
			367E.
			367F.
			367G. Corollary
			367H. Proposition
			367I. Dominated convergence
			367J. The Martingale Theorem
			367K.
			367L.
			367M. Theorem
			367N. Proposition
			367O. Theorem
			367P. Proposition
			367Q.
			367R.
			367S. Proposition
			367T. Intrinsic description of convergence in measure
			*367U. Theorem
			*367V. Corollary
			*367W. Independence
			367X. Basic exercises
			367Y. Further exercises
			367 Notes and comments
		368. Embedding Riesz spaces in L^0
			368A. Lemma
			368B. Theorem
			368C. Corollary
			368D. Corollary
			368E. Theorem
			368F. Corollary
			368G. Corollary
			368H. Corollary
			368I. Corollary
			368J. Definition
			368K.
			368L. Definition
			368M. Theorem
			368N Weakly (σ, ∞)-distributive Riesz spaces
			368O. Lemma
			368P. Proposition
			368Q. Theorem
			368R. Corollary
			368S. Corollary
			368X. Basic exercises
			368Y. Further exercises
			368 Notes and comments
		369. Banach function spaces
			369A. Theorem
			369B. Corollary
			369C.
			369D. Corollary
			369E. Kakutani's theorem
			369F.
			369G. Proposition
			369H. Associate norms
			369I. Theorem
			369J. Theorem
			369K. Corollary
			369L. L^p
			369M. Proposition
			369N.
			369O. Proposition
			369P.
			369Q. Corollary
			369R.
			369X. Basic exercises
			369Y. Further exercises
			369 Notes and comments
	Chapter 37. Linear operators between function spaces
		371. The Chacon-Krengel theorem
			371A. Lemma
			371B. Theorem
			371C. Theorem
			371D. Corollary
			371E. Remarks
			371F. The class T^{(0)}
			371G. Proposition
			371H. Remark
			371X. Basic exercises
			371Y. Further exercises
			371 Notes and comments
		372. The ergodic theorem
			372A. Lemma
			372B. Lemma
			372C. Maximal Ergodic Theorem
			372D.
			372E. Corollary
			372F. The Ergodic Theorem: second form
			372G. Corollary
			372H.
			372I.
			372J. The Ergodic Theorem: third form
			372K. Remark
			372L. Continued fractions
			372M. Theorem
			372N. Corollary
			372O. Mixing and ergodic transformations
			372P.
			372Q.
			372R. Remarks
			372S.
			372X. Basic exercises
			372Y. Further exercises
			372 Notes and comments
		373. Decreasing rearrangements
			373A. Definition
			373B. Proposition
			373C. Decreasing rearrangements
			373D. Lemma
			373E. Theorem
			373F. Theorem
			373G. Lemma
			373H. Lemma
			373I. Lemma
			373J. Corollary
			373K. The very weak operator topology of T
			373L. Theorem
			373M. Corollary
			373N. Corollary
			373O. Theorem
			373P. Theorem
			373Q. Corollary
			373R. Order-continuous operators: Proposition
			373S. Adjoints in T^{(0)}
			373T. Corollary
			373U. Corollary
			373X. Basic exercises
			373Y. Further exercises
			373 Notes and comments
		374. Rearrangement-invariant spaces
			374A. T-invariance
			374B.
			374C.
			374D.
			374E.
			374F. Remarks
			374G. Definition
			374H. Proposition
			374I. Corollary
			374J. Lemma
			374K. Theorem
			374L. Lemma
			374M. Proposition
			374X. Basic exercises
			374Y. Further exercises
			374 Notes and comments
		375. Kwapien's theorem
			375A. Theorem
			375B. Proposition
			375C. Theorem
			375D. Corollary
			375E. Theorem
			375F.
			375G. Lemma
			375H. Lemma
			375I. Lemma
			375J. Theorem
			375K. Corollary
			375L. Corollary
			375X. Basic exercises
			375Y. Further exercises
			375Z. Problem
			375 Notes and comments
		376. Kernel operators
			376A. Kernel operators
			376B. The canonical map L^0 × L^0 -> L^0
			376C.
			376D. Abstract integral operators
			376E. Theorem
			376F. Corollary
			376G. Lemma
			376H. Theorem
			376I.
			376J. Corollary
			376K.
			376L. Lemma
			376M. Theorem
			376N. Corollary: Dunford's theorem
			376O.
			376P. Theorem
			376Q. Corollary
			376R.
			376S. Theorem
			376X. Basic exercises
			376Y. Further exercises
			376 Notes and comments
		*377 .Function spaces of reduced products
			377A. Proposition
			377B. Theorem
			377C. Theorem
			377D.
			377E. Proposition
			377F.
			377G. Projective limits
			377H. Inductive limits
			377X. Basic exercises
			377Y. Further exercises
			377 Notes and comments
	Chapter 38. Automorphism groups
		381. Automorphisms of Boolean algebras
			381A. The group Aut \mathfrak{A}
			381B.
			381C.
			381D. Corollary
			381E. Lemma
			381F. Corollary
			381G. Corollary
			381H. Proposition
			381I. Full and countably full subgroups
			381J. Lemma
			381K. Lemma
			381L. Lemma
			381M.
			381N. Lemma
			381O. Lemma
			381P. Proposition
			381Q.
			381R. Cyclic automorphisms
			381S. Lemma
			381X. Basic exercises
			381Y. Further exercises
		382. Factorization of automorphisms
			382A. Definitions
			382B. Lemma
			382C. Corollary
			382D. Lemma
			382E. Corollary
			382F. Corollary
			382G. Lemma
			382H. Lemma
			382I. Lemma
			382J. Lemma
			382K. Lemma
			382L. Lemma
			382M. Theorem
			382N. Corollary
			382O. Definition
			382P. Lemma
			382Q. Lemma
			382R. Theorem
			382S. Corollary
			382X. Basic exercises
			382Y. Further exercises
			382 Notes and comments
		383. Automorphism groups of measure algebras
			383A. Definition
			383B. Lemma
			383C. Corollary
			383D. Theorem
			383E. Lemma
			383F. Lemma
			383G. Lemma
			383H. Corollary
			383I. Normal subgroups of Aut \mathfrak{A} and Aut_{\bar{\mu}} \mathfrak{A}
			383J.
			383K.
			383L. Corollary
			383X. Basic exercises
			383Y. Further exercises
			383 Notes and comments
		384. Outer automorphisms
			384A. Lemma
			384B. A note on supports
			384C. Lemma
			384D. Theorem
			384E.
			384F. Corollary
			384G. Corollary
			384H. Definitions
			384I. Lemma
			384J. Theorem
			384K. Corollary
			384L. Examples
			384M. Theorem
			384N.
			384O. Corollary
			384P.
			384Q. Example
			384X. Basic exercises
			384Y. Further exercises
			384 Notes and comments
		385. Entropy
			385A. Notation
			385B. Lemma
			385C. Definition
			385D. Definition
			385E. Elementary remarks
			385F. Definition
			385G. Lemma
			385H. Corollary
			385I. Lemma
			385J. Lemma
			385K. Definition
			385L. Lemma
			385M. Definition
			385N. Lemma
			385O. Lemma
			385P. Theorem (Kolmogorov 58, Sinai 59)
			385Q. Bernoulli shifts
			385R. Theorem
			385S. Remarks
			385T. Isomorphic homomorphisms
			385U. Definition
			385V.
			385X. Basic exercises
			385Y. Further exercises
			385 Notes and comments
		386. More about entropy
			386A.
			386B. Corollary
			386C. The Halmos-Rokhlin-Kakutani lemma
			386D. Corollary
			386E.
			386F. Corollary
			386G. Definition
			386H. Lemma
			386I. Corollary
			386J.
			386K. Lemma
			386L. Lemma
			386M. Lemma
			386N. Lemma
			386O. Lemma
			386X. Basic exercises
			386Y. Further exercises
			386 Notes and comments
		387. Ornstein's theorem
			387A.
			387B. Remarks
			387C. Lemma
			387D. Corollary
			387E. Sinaĭ's theorem (atomic case) (SINAĬ 62)
			387F. Lemma
			387G. Lemma
			387H. Lemma
			387I. Ornstein's theorem (finite entropy case)
			387J.
			387K. Ornstein's theorem (infinite entropy case)
			387L. Corollary: Sinaĭ's theorem (general case)
			387X. Basic exercises
			387Y. Further exercises
			387 Notes and comments
		388. Dye's theorem
			388A. Orbit structures
			388B. Corollary
			388C.
			388D. von Neumann automorphisms
			388E. Example
			388F.
			388G. Lemma
			388H. Lemma
			388I. Lemma
			388J. Lemma
			388K. Theorem
			388L. Theorem
			388X. Basic exercises
			388Y. Further exercises
			388 Notes and comments
	Chapter 39. Measurable algebras
		391. Kelley's theorem
			391A. Proposition
			391B. Definition
			391C. Proposition
			391D. Theorem (Kantorovich Vulikh & Pinsker 50)
			391E.
			391F. Theorem
			391G. Corollary
			391H. Definition
			391I. Proposition
			391J. Theorem
			391K. Corollary
			391X. Basic exercises
			391Y. Further exercises
			391 Notes and comments
		392. Submeasures
			392A. Definition
			392B.
			392C. Proposition
			392D. Lemma
			392E. Lemma
			392F. Theorem
			392G. Corollary
			392H.
			392I. Corollary
			392J. Proposition
			*392K. Products of submeasures
			392X. Basic exercises
			392Y. Further exercises
			392 Notes and comments
		393. Maharam submeasures
			393A. Definition
			393B. Lemma
			393C. Proposition
			393D. Theorem
			393E. Maharam algebras
			393F. Lemma
			393G. Proposition
			393H. Proposition
			393I. Proposition
			393J. Lemma
			*393K. Theorem
			393L.
			393M. Lemma
			393N. Proposition
			393O. Proposition
			393P. Lemma
			393Q. Theorem (Balcar Głowczynski & Jech 98, Balcar Jech & Pazák 05)
			393R. Definition
			393S. Theorem (TODORČEVIĆ 04)
			393X. Basic exercises
			393Y. Further exercises
			393 Notes and comments
		394. Talagrand's example
			394A.
			394B. Lemma
			394C. Definitions
			394D. Very elementary facts
			394E. Lemma
			394F. Corollary
			394G.
			394H. Definitions
			394I. Proposition
			394J. Lemma
			394K. Lemma
			394L. Lemma
			394M. Theorem
			394N. Remarks
			*394O. Control measures
			*394P. Example
			*394Q.
			394X. Basic exercises
			394Y. Further exercises
			394Z. Problems
			394 Notes and comments
		395. Kawada's theorem
			395A. Definitions
			395B.
			395C. Lemma
			395D. Theorem
			395E. Definition
			395F. Proposition
			395G. The fixed-point subalgebra of a group
			395H.
			395I.
			395J. Notation
			395K. Lemma
			395L. Lemma
			395M. Lemma
			395N.
			395O.
			395P. Theorem
			395Q. Corollary: Kawada's theorem
			395R.
			395X. Basic exercises
			395Y. Further exercises
			395Z. Problem
			395 Notes and comments
		396. The Hajian-Ito theorem
			396A. Lemma
			396B. Theorem (Hajian & Ito 69)
			396C. Remark
			396X. Basic exercises
			396Y. Further exercises
			396 Notes and comments
	Appendix to Volume 3 - Useful Facts
		3A1. Set Theory
			3A1A. The axioms of set theory
			3A1B. Definition
			3A1C. Calculation of cardinalities
			3A1D. Cardinal exponentiation
			3A1E. Definition
			3A1F. Cofinal sets
			3A1G. Zorn's Lemma
			3A1H. Natural numbers and finite ordinals
			3A1I. Definitions
			3A1J. Subsets of given size
			3A1K.
		3A2. Rings
			3A2A. Definition
			3A2B. Elementary facts
			3A2C. Subrings
			3A2D. Homomorphisms
			3A2E. Ideals
			3A2F. Quotient rings
			3A2G. Factoring homomorphisms through quotient rings
			3A2H. Product rings
		3A3. General topology
			3A3A. Taxonomy of topological spaces
			3A3B. Elementary relationships
			3A3C. Continuous functions
			3A3D. Compact spaces
			3A3E. Dense sets
			3A3F. Meager sets
			3A3G. Baire's theorem for locally compact Hausdorff spaces
			3A3H. Corollary
			3A3I. Product spaces
			3A3J. Tychonoff's theorem
			3A3K. The spaces {0, 1}^{I}
			3A3L. Cluster points of filters
			3A3M. Topology bases
			3A3N. Uniform convergence
			3A3O. One-point compactifications
			3A3P. Topologies defined from a sequential convergence
			3A3Q. Miscellaneous definitions
		3A4. Uniformities
			3A4A. Uniformities
			3A4B. Uniformities and pseudometrics
			3A4C. Uniform continuity
			3A4D. Subspaces
			3A4E. Product uniformities
			3A4F. Completeness
			3A4G. Extension of uniformly continuous functions
			3A4H. Completions
			3A4I. A note on metric spaces
		3A5. Normed spaces
			3A5A. The Hahn-Banach theorem
			3A5B. Cones
			3A5C. Hahn-Banach theorem: geometric forms
			3A5D. Separation from finitely-generated cones
			3A5E. Weak topologies
			3A5F. Weak* topologies
			3A5G. Reflexive spaces
			3A5H. Uniform Boundedness Theorem
			*3A5I. Strong operator topologies
			3A5J. Completions
			3A5K. Normed algebras
			3A5L. Compact operators
			3A5M. Hilbert spaces
			*3A5N. Bounded sets in linear topological spaces
		3A6. Group Theory
			3A6A. Definition
			3A6B. Definition
			3A6C. Normal subgroups
	Concordance
	References for Volume 3
	Index to volumes 1, 2 and 3
		Principal topics and results
		General index
Measure Theory 4-1_Topological Measure Spaces(2e,2013,577p)D.H.Fremlin_9780956607126
	Contents
		General introduction
		Introduction to Volume 4
		Note on second printing
		Note on second (`Lulu') edition
	Chapter 41. Topologies and Measures I
		411. Definitions
			411A.
			411B.
			411C. Definition
			411D.
			411E.
			411F.
			411G. Elementary facts
			411H.
			411I. Remarks
			411J.
			411K. Borel and Baire measures
			411L.
			411M. Definition
			411N.
			411O. Example
			411P. Example: Stone spaces
			411Q. Example: Dieudonné's measure
			411R. Example: The Baire σ-algebra of ω_1
			411X. Basic exercises
			411Y. Further exercises
			411 Notes and comments
		412. Inner regularity
			412A.
			412B. Corollary
			412C.
			412D.
			412E. Theorem
			412F. Lemma
			412G. Theorem
			412H. Proposition
			412I. Lemma
			412J. Proposition
			412K. Proposition
			412L. Corollary
			412M. Corollary
			412N. Lemma
			412O. Lemma
			412P. Proposition
			412Q. Proposition
			412R. Lemma
			412S. Proposition
			412T. Lemma
			412U. Proposition
			412V. Corollary
			*412W. Outer regularity
			412X. Basic exercises
			412Y. Further exercises
			412 Notes and comments
		413. Inner measure constructions
			413A.
			413B.
			413C. Measures from inner measures
			413D. The inner measure defined by a measure
			413E.
			413F.
			413G.
			413H.
			413I. Theorem (Topsøe 70A)
			413J. Theorem
			413K. Corollary
			413L.
			413M. Corollary
			413N.
			413O. Corollary
			413P.
			413Q. Theorem
			413R.
			413S. Corollary
			413T.
			413X. Basic exercises
			413Y. Further exercises
			413 Notes and comments
		414. τ-additivity
			414A. Theorem
			414B. Corollary
			414C. Corollary
			414D. Corollary
			414E. Corollary
			414F. Corollary
			414G. Corollary
			414H. Corollary
			414I. Proposition
			414J. Theorem
			414K. Proposition
			414L. Lemma
			414M. Proposition
			414N. Proposition
			414O.
			414P. Density topologies
			414Q. Lifting topologies
			414R. Proposition
			414X. Basic exercises
			414Y. Further exercises
			414 Notes and comments
		415. Quasi-Radon measure spaces
			415A. Theorem
			415B. Theorem
			415C.
			415D.
			415E.
			415F. Corollary
			415G. Comparing quasi-Radon measures
			415H. Uniqueness of quasi-Radon measures
			415I. Proposition
			415J. Proposition
			415K.
			415L. Proposition
			415M. Corollary
			415N. Corollary
			415O. Proposition
			415P. Proposition
			415Q.
			415R. Proposition
			415X. Basic exercises
			415Y. Further exercises
			415 Notes and comments
		416. Radon measure spaces
			416A. Proposition
			416B. Corollary
			416C.
			416D.
			416E. Specification of Radon measures
			416F. Proposition
			416G.
			416H. Corollary
			416I.
			416J.
			416K. Proposition (see TOPSØE 70A)
			416L. Proposition
			416M. Corollary
			416N. Henry's theorem (Henry 69)
			416O. Theorem
			416P. Theorem
			416Q. Proposition
			416R. Theorem
			416S.
			416T.
			416U. Theorem
			416V. Stone spaces
			416W. Compact measure spaces
			416X. Basic exercises
			416Y. Further exercises
			416 Notes and comments
		417. τ-additive product measures
			417A. Lemma
			417B. Lemma
			417C. Theorem (RESSEL 77)
			417D. Multiple products
			417E. Theorem
			417F. Corollary
			417G. Notation
			417H. Fubini's theorem for τ-additive product measures
			417I.
			417J.
			417K. Proposition
			417L. Corollary
			417M. Proposition
			417N. Theorem
			417O. Theorem
			417P. Theorem
			417Q. Theorem
			417R. Notation
			417S.
			417T. Proposition
			417U. Proposition
			417V. Proposition
			417X. Basic exercises
			417Y. Further exercises
			417 Notes and comments
		418. Measurable functions and almost continuous functions
			418A. Proposition
			418B. Proposition
			418C. Proposition
			418D. Proposition
			418E. Theorem
			418F. Proposition
			418G. Proposition
			418H. Proposition
			418I.
			418J. Theorem
			418K. Corollary
			418L.
			418M. Prokhorov's theorem
			418N. Remarks
			418O.
			418P. Proposition
			418Q. Corollary
			418R.
			418S. Corollary
			418T. Corollary (MAULDIN & STONE 81)
			*418U. Independent families of measurable functions
			418X. Basic exercises
			418Y. Further exercises
			418 Notes and comments
		419. Examples
			419A. Example
			419B. Lemma
			419C. Example (FREMLIN 75B)
			419D. Example (FREMLIN 75B)
			419E. Example (FREMLIN 76)
			419F. Theorem (RAO 69)
			419G. Corollary (ULAM 30)
			419I.
			419J. Example
			419K. Example (BLACKWELL 56)
			419L. The split interval again
			419X. Basic exercises
			419Y. Further exercises
			419 Notes and comments
	Chapter 42. Descriptive set theory
		421. Souslin's operation
			421A. Notation
			421B. Definition
			421C. Elementary facts
			421D.
			421E. Corollary
			421F. Corollary
			421G. Proposition
			421H.
			421I.
			421J. Proposition
			421K. Definition
			421L. Proposition
			421M. Proposition
			*421N.
			*421O. Theorem
			*421P. Corollary
			*421Q. Lemma
			421X. Basic exercises
			421Y. Further exercises
			421 Notes and comments
		422. K-analytic spaces
			422A. Definition
			422B.
			422C. Proposition
			422D. Lemma
			422E.
			422F Definition (FROLÍK 61)
			422G. Theorem
			422H. Theorem
			422I.
			422J. Corollary
			*422K.
			422X. Basic exercises
			422Y. Further exercises
			422 Notes and comments
		423. Analytic spaces
			423A. Definition
			423B. Proposition
			423C. Theorem
			423D. Corollary
			423E. Theorem
			423F. Proposition
			423G. Lemma
			423H. Lemma
			423I. Theorem
			423J. Lemma
			423K. Corollary
			423L. Proposition
			423M.
			423N.
			423O. Corollary
			*423P. Constituents of coanalytic sets
			*423Q. Remarks
			*423R. Coanalytic and PCA sets
			423S. Proposition
			423X. Basic exercises
			423Y. Further exercises
			423 Notes and comments
		424. Standard Borel spaces
			424A. Definition
			424B. Proposition
			424C. Theorem
			424D. Corollary
			424E. Proposition
			424F. Corollary
			424G. Proposition
			*424H.
			424X. Basic exercises
			424Y. Further exercises
			424 Notes and comments
		*425. Realization of automorphisms
			425A.
			425B. Lemma
			425C. Master actions
			425D. Törnquist's theorem (TÖRNQUIST 11)
			425E. Scholium
			425X. Basic exercises
			425Y. Further exercises
			425Z. Problems
			425 Notes and comments
	Chapter 43. Topologies and measures II
		431. Souslin's operation
			431A. Theorem
			431B. Corollary
			431C. Corollary
			431D. Theorem
			431E. Corollary
			*431F.
			*431G.
			431X. Basic exercises
			431Y. Further exercises
			431 Notes and comments
		432. K-analytic spaces
			432A. Proposition
			432B. Theorem
			432C. Proposition
			432D. Theorem (ALDAZ & RENDER 00)
			432E. Corollary
			432F. Corollary
			432G. Corollary
			432H. Corollary
			432I. Corollary
			432J. Capacitability
			432K. Theorem (CHOQUET 55)
			432L. Proposition
			432X. Basic exercises
			432Y. Further exercises
			432 Notes and comments
		433. Analytic spaces
			433A. Proposition
			433B. Lemma
			433C. Theorem
			433D. Theorem
			433E. Proposition
			433F.
			433G. Proposition
			433H. Proposition
			433I.
			433J. Proposition
			433K.
			433L. Proposition
			433X. Basic exercises
			433Y. Further exercises
			433 Notes and comments
		434. Borel measures
			434A. Types of Borel measures
			434B. Compact, analytic and K-analytic spaces
			434C. Radon spaces
			434D. Universally measurable sets
			434E. Universally Radon-measurable sets
			434F. Elementary properties of Radon spaces
			434G.
			434H. Proposition
			434I. Proposition
			434J. Proposition
			434K.
			434L.
			434M.
			434N. Proposition
			434O. Quasi-dyadic spaces
			434P. Proposition
			434Q. Theorem (FREMLIN & GREKAS 95)
			434R.
			*434S.
			*434T.
			434X. Basic exercises
			434Y. Further exercises
			434Z. Problems
			434 Notes and comments
		435. Baire measures
			435A. Types of Baire measures
			435B. Theorem
			435C. Theorem (MARÍK 57)
			435D.
			435E.
			435F. Elementary facts
			435G. Proposition
			435H. Corollary
			435X. Basic exercises
			435Y. Further exercises
			435 Notes and comments
		436. Representation of linear functionals
			436A. Definition
			436B. Definition
			436C. Lemma
			436D. Theorem
			436E. Proposition
			436F.
			436G. Definition
			436H. Theorem
			436I. Lemma
			436J. Riesz Representation Theorem (first form)
			436K. Riesz Representation Theorem (second form)
			*436L.
			*436M. Corollary
			436X. Basic exercises
			436Y. Further exercises
			436 Notes and comments
		437. Spaces of measures
			437A. Smooth and sequentially smooth duals
			437B. Signed measures
			437C. Theorem
			437D. Remarks
			437E. Corollary
			437F. Proposition
			437G. Definitions
			437H. Theorem
			437I. Proposition
			437J. Vague and narrow topologies
			437K. Proposition
			437L. Corollary
			437M. Theorem (RESSEL 77)
			437N.
			437O. Uniform tightness
			437P. Proposition
			437Q. Two metrics
			437R. Theorem
			437S.
			437T.
			437U.
			437V. Theorem
			437X. Basic exercises
			437Y. Further exercises
			437 Notes and comments
		438. Measure-free cardinals
			438A. Measure-free cardinals
			438B.
			438C.
			438D.
			438E. Proposition
			438F. Proposition
			438G. Corollary
			438H.
			438I. Proposition
			438J.
			438K. Hereditarily weakly θ-refinable spaces
			438L. Lemma
			438M. Proposition (GARDNER 75)
			438N.
			438O. Lemma
			438P. Lemma
			438Q. Theorem
			438R. Corollary
			*438S. Càllàl functions
			438T. Proposition
			438U.
			438X. Basic exercises
			438Y. Further exercises
			438 Notes and comments
		439. Examples
			439A. Example
			439B. Definition
			439C. Proposition
			439D. Remarks
			439E. Lemma
			439F. Proposition
			439G. Corollary
			439H. Corollary
			439I. Example
			439J. Example
			439K. Example
			439L. Example
			439M. Example
			439N. Example
			439O.
			439P. Example (cf. MORAN 68)
			439Q. Example
			439R. Example
			439S.
			439X. Basic exercises
			439Y. Further exercises
			439 Notes and comments
	Chapter 44. Topological groups
		441. Invariant measures on locally compact spaces
			441A. Group actions
			441B.
			441C. Theorem (STEINLAGE 75)
			441D.
			441E. Theorem
			441F.
			441G. The topology of an isometry group
			441H. Theorem
			441I. Remarks
			441J.
			441K. Theorem
			1L P.roposition
			441X. Basic exercises
			441Y. Further exercises
			441 Notes and comments
		442. Uniqueness of Haar measures
			442A. Lemma
			442B. Theorem
			442C. Proposition
			442D. Remark
			442E. Lemma
			442F. Domains of Haar measures
			442G. Corollary
			442H. Remark
			442I. The modular function
			442J. Proposition
			442K. Theorem
			442L. Corollary
			442X. Basic exercises
			442Y. Further exercises
			442Z. Problem
			442 Notes and comments
		443. Further properties of Haar measure
			443A. Haar measurability
			443B. Lemma
			443C. Theorem
			443D. Proposition
			443E. Corollary
			443F.
			443G.
			443H. Theorem
			443I. Corollary
			443J. Proposition
			443K. Theorem
			443L. Corollary
			443M. Theorem (HALMOS 50)
			443N.
			443O.
			443P. Quotient spaces
			443Q. Theorem
			443R. Theorem
			443S. Applications
			443T. Theorem
			443U. Transitive actions
			443X. Basic exercises
			443Y. Further exercises
			443 Notes and comments
		444. Convolutions
			444A. Convolution of measures
			444B. Proposition
			444C. Theorem
			444D. Proposition
			444E. The Banach algebra of τ-additive measures
			444F.
			444G. Corollary
			444H. Convolutions of measures and functions
			444I. Proposition
			444J. Convolutions of functions and measures
			444K. Proposition
			444L. Corollary
			444M. Proposition
			444N.
			444O. Convolutions of functions
			444P. Proposition
			444Q. Proposition
			444R. Proposition
			444S. Remarks
			444T. Proposition
			444U. Corollary
			444V.
			444X. Basic exercises
			444Y .Further exercises
			444 Notes and comments
		445. The duality theorem
			445A. Dual groups
			445B. Examples
			445C. Fourier-Stieltjes transforms
			445D. Theorem
			445E.
			445F. Fourier transforms of functions
			445G. Proposition
			445H. Theorem
			445I. The topology of the dual group
			445J. Corollary
			445K. Proposition
			445L. Positive definite functions
			445M. Proposition
			445N. Bochner's theorem (HERGLOTZ 1911, BOCHNER 33, WEIL 40)
			445O. Proposition
			445P. The Inversion Theorem
			445Q. Remark
			445R. The Plancherel Theorem
			445S.
			445T. Corollary
			445U. The Duality Theorem (PONTRYAGIN 34, KAMPEN 35)
			445X. Basic exercises
			445Y. Further exercises
			445 Notes and comments
		446. The structure of locally compact groups
			446A. Finite-dimensional representations
			446B. Theorem
			446C. Corollary
			*446D. Notation
			*446E. Lemma
			*446F. Lemma
			*446G. `Groups with no small subgroups'
			*446H. Lemma
			*446I. Lemma
			*446J. Lemma
			*446K. Lemma
			*446L. Definition
			*446M. Proposition
			*446N. Proposition
			*446O. Theorem
			*446P. Corollary
			446X. Basic exercises
			446Y. Further exercises
			446 Notes and comments
		447. Translation-invariant liftings
			447A. Liftings and lower densities
			447B. Lemma
			447C. Vitali's theorem
			447D. Theorem
			447E.
			447F. Lemma
			447G. Lemma
			447H. Lemma
			447I. Theorem (IONESCU TULCEA & IONESCU TULCEA 67)
			447J. Corollary
			447X. Basic exercises
			447Y. Further exercises
			447 Notes and comments
		448. Polish group actions
			448A. Definitions
			448B.
			448C. Lemma
			448D. Theorem
			448E. Definition
			448F.
			448G.
			448H. Lemma
			448I. Notation
			448K.
			448L.
			448M. Lemma
			448N. Theorem
			448O.
			448P.
			448Q.
			448R. Lemma
			448S. Mackey's theorem (MACKEY 62)
			448T. Corollary
			448X. Basic exercises
			448Y. Further exercises
			448 Notes and comments
		449. Amenable groups
			449A. Definition
			449B. Lemma
			449C. Theorem
			449D. Theorem
			449E. Corollary
			449F. Corollary
			449G. Example
			449H.
			449I. Notation
			449J. Theorem
			449K. Proposition
			449L.
			449M. Corollary
			449N. Theorem
			449O. Corollary (BANACH 1923)
			449X. Basic exercises
			449Y. Further exercises
			449 Notes and comments
	Chapter 45. Perfect measures and disintegrations
		451. Perfect, compact and countably compact measures
			451A.
			451B.
			451C. Proposition (RYLL-NARDZEWSKI 53)
			451D. Proposition
			451E. Proposition
			451F. Lemma (SAZONOV 66)
			451G. Proposition
			451H. Lemma
			451I. Theorem
			451J. Theorem
			451K.
			*451L.
			451M.
			451N. Proposition
			451O. Corollary
			451P. Corollary
			451Q.
			451R. Lemma
			451S. Proposition
			451T. Theorem (FREMLIN 81, KOUMOULLIS & PRIKRY 83)
			451U. Example (VINOKUROV & MAKHKAMOV 73, MUSIAŁ 76)
			*451V Weakly α-favourable spaces
			451X. Basic exercises
			451Y. Further exercises
			451 Notes and comments
		452. Integration and disintegration of measures
			452A. Lemma
			452B. Theorem
			452C. Theorem
			452D. Theorem
			452E.
			452F. Proposition
			452G.
			452H. Lemma
			452I. Theorem (PACHL 78)
			452J. Remarks
			452K. Example
			452L.
			452M.
			452N. Corollary
			452O. Proposition
			452P. Corollary (cf. BLACKWELL 56)
			452Q. Disintegrations and conditional expectations
			*452R.
			*452S. Corollary (PACHL 78)
			452T.
			452X. Basic exercises
			452Y. Further exercises
			452 Notes and comments
		453. Strong liftings
			453A.
			453B.
			453C. Proposition
			453D. Proposition
			453E. Proposition
			453F. Proposition
			453G. Corollary
			453H. Lemma
			453I. Proposition
			453J. Corollary
			453K.
			453L. Remark
			453M. Strong liftings and Stone spaces
			453N. Losert's example (LOSERT 79)
			453X. Basic exercises
			453Y. Further exercises
			453Z. Problems
			453 Notes and comments
		454. Measures on product spaces
			454A. Theorem
			454B. Corollary
			454C. Theorem (MARCZEWSKI & RYLL-NARDZEWSKI 53)
			454D. Theorem (KOLMOGOROV 33, §III.4)
			454E. Corollary
			454F. Corollary
			454G. Corollary
			454H. Corollary
			454I. Remarks
			454J. Distributions of random processes
			454K. Definition
			454L. Independence
			454M.
			454N.
			454O. Proposition
			454P. Theorem
			454Q. Continuous processes
			454R. Proposition
			454S. Corollary
			454T.
			454X. Basic exercises
			454Y. Further exercises
			454 Notes and comments
		455. Markov and Lévy processes
			455A. Theorem
			455B. Lemma
			455C. Theorem
			455D. Remarks
			455E. Theorem
			455F.
			455G. Theorem
			455H. Corollary
			455I.
			455J. Theorem
			455K. Corollary
			455L. Stopping times
			455M. Hitting times
			455N.
			455O.
			455P.
			455Q. Lévy processes
			455R. Theorem
			455S. Lemma
			455T. Corollary
			455U. Theorem
			455X. Basic exercises
			455Y. Further exercises
			455 Notes and comments
		456. Gaussian distributions
			456A. Definitions
			456B.
			456C.
			456D. Gaussian processes
			456E. Independence and correlation
			456F. Proposition
			456G.
			456H. The support of a Gaussian distribution
			456I. Remarks
			456J. Universal Gaussian distributions
			456K. Proposition
			456L. Lemma
			456M. Cluster sets
			456N. Lemma
			456O.
			456P. Corollary
			456Q. Proposition
			456X. Basic exercises
			456Y. Further exercises
			456 Notes and comments
		457. Simultaneous extension of measures
			457A.
			457B. Corollary
			457C. Corollary
			*457D.
			457E. Proposition
			457F. Proposition
			457G. Theorem
			457H. Example
			457I. Example
			457J. Example
			457K.
			457L. Theorem
			457M.
			457N. Remarks
			457X. Basic exercises
			457Y. Further exercises
			457Z. Problems
			457 Notes and comments
		458. Relative independence and relative products
			458A. Relative independence
			458B.
			458C. Proposition
			458D. Proposition
			458E. Example
			458F.
			*458G.
			458H.
			458I.
			458J. Theorem
			458K.
			458L. Measure algebras
			458M. Proposition
			458N. Relative free products of probability algebras
			458O. Theorem
			458P.
			458Q. Relative product measures
			458R. Proposition
			458S.
			458T.
			458U.
			458X. Basic exercises
			458Y. Further exercises
			458 Notes and comments
		459. Symmetric measures and exchangeable random variables
			459A.
			459B. Theorem
			459C. Exchangeable random variables
			459D.
			459E.
			459F. Lemma
			459G. Lemma
			459H. Theorem
			459I.
			459J. Corollary
			459K.
			459X. Basic exercises
			459Y. Further exercises
			459 Notes and comments
	Concordance to chapters 41-45
	References
Measure Theory 4-2_Topological Measure Spaces(2e,2013,573p)D.H.Fremlin_9780956607133
	Contents
	Chapter 46. Pointwise compact sets of measurable functions
		461. Barycenters and Choquet's theorem
			461A. Definitions
			461B. Proposition
			461C. Lemma
			461D. Theorem
			461E. Theorem
			461F. Theorem
			461G. Lemma
			461H. Proposition
			461I. Theorem
			461J. Corollary: Kreĭn's theorem
			461K. Lemma
			461L. Lemma
			461M. Theorem
			461N. Lemma
			461O. Lemma
			461P. Theorem
			461Q.
			461R. Corollary
			461X. Basic exercises
			461Y. Further exercises
			461 Notes and comments
		462. Pointwise compact sets of continuous functions
			462A. Definitions
			*462B. Proposition (PRYCE 71)
			*462C. Theorem (PRYCE 71)
			*462D. Theorem
			462E. Theorem
			462F. Lemma
			462G. Proposition
			462H. Lemma
			462I. Theorem
			462J. Corollary
			462K. Proposition
			462L. Corollary
			462X. Basic exercises
			462Y. Further exercises
			462Z. Problem
			462 Notes and comments
		463. \mathfrak{T}_p and \mathfrak{T}_m
			463A. Preliminaries
			463B. Lemma
			463C. Proposition (IONESCU TULCEA 73)
			463D. Lemma
			463E. Proposition
			463F. Corollary
			463G. Theorem (IONESCU TULCEA 74)
			463H. Corollary
			463I. Lemma
			463J. Lemma
			463K. Fremlin's Alternative (FREMLIN 75A)
			463L. Corollary
			463M. Proposition
			463N. Corollary
			463X. Basic exercises
			463Y. Further exercises
			463Z. Problems
			463 Notes and comments
		464. Talagrand's measure
			464A. The usual measure on PI
			464B. Lemma
			464C. Lemma
			464D. Construction (TALAGRAND 80)
			464E. Example
			464F. The L-space \ell^\infty(I)^\ast
			464G.
			464H.
			464I. Measurable and purely non-measurable functionals
			464J. Examples
			464K. The space M_m
			464L. The space M_{pnm}
			464M. Theorem (FREMLIN & TALAGRAND 79)
			464N. Corollary (FREMLIN & TALAGRAND 79)
			464O. Remark
			464P. More on purely non-measurable functionals
			464Q. More on measurable functionals
			464R. A note on  \ell^\infty(I)
			464X. Basic exercises
			464Y. Further exercises
			464Z. Problem
			464 Notes and comments
		465. Stable sets
			465A. Notation
			465B. Definition
			465C.
			465D.
			465E. The topology \mathfrak{T}_s(\mathcal{L}^2, \mathcal{L}^2)
			465F. Lemma
			465G. Theorem
			465H.
			465I.
			465J.
			465K. Lemma
			465L. Lemma (TALAGRAND 87)
			465M. Theorem (TALAGRAND 82, TALAGRAND 87)
			465N. Theorem
			465O. Stable sets in L^0
			465P. Theorem
			465Q. Remarks
			465R. Theorem (TALAGRAND 84)
			*465S. R-stable sets
			*465T. Proposition (TALAGRAND 84)
			*465U.
			*465V. Remark
			465X. Basic exercises
			465Y. Further exercises
			465 Notes and comments
		466. Measures on linear topological spaces
			466A. Theorem
			466B. Corollary
			466C. Definition
			466D. Proposition (HANSELL 01)
			466E. Corollary
			466F. Proposition
			466G. Definition
			466H. Proposition (JAYNE & ROGERS 95)
			466I. Examples
			466J. Theorem
			466K. Proposition
			466L. Proposition
			466M. Corollary
			466N. Gaussian measures
			466O. Proposition
			466X. Basic exercises
			466Y. Further exercises
			466Z. Problems
			466 Notes and comments
		*467 .Locally uniformly rotund norms
			467A. Definition
			467B. Proposition
			467C. A technical device
			467D. Lemma
			467E. Theorem
			467F. Lemma
			467G. Theorem
			467H. Definitions
			467I. Lemma
			467J. Lemma
			467K. Theorem
			467L. Weakly compactly generated Banach spaces
			467M. Proposition (TALAGRAND 75)
			467N. Theorem
			467O. Eberlein compacta
			467P. Proposition
			467X. Basic exercises
			467Y. Further exercises
			467 Notes and comments
	Chapter 47. Geometric measure theory
		471. Hausdorff measures
			471A. Definition
			471B. Definition
			471C. Proposition
			471D. Theorem
			471E. Corollary
			471F. Corollary
			471G. Increasing Sets Lemma (DAVIES 70)
			471H. Corollary
			471I. Theorem
			471J. Proposition
			471K. Lemma
			471L. Proposition
			471M.
			471N. Lemma
			471O. Lemma
			471P. Theorem
			471Q.
			471R. Lemma (HOWROYD 95)
			471S. Theorem (HOWROYD 95)
			471T. Proposition
			471X. Basic exercises
			471Y. Further exercises
			471 Notes and comments
		472. Besicovitch's Density Theorem
			472A. Besicovitch's Covering Lemma
			472B. Theorem
			472C. Theorem
			472D. Besicovitch's Density Theorem
			*472E. Proposition
			*472F. Theorem
			472X. Basic exercises
			472Y. Further exercises
			472 Notes and comments
		473. Poincaré's inequality
			473A. Notation
			473B. Differentiable functions
			473C. Lipschitz functions
			473D. Smoothing by convolution
			473E. Lemma
			473F. Lemma
			473G. Lemma
			473H. Gagliardo-Nirenberg-Sobolev inequality
			473I. Lemma
			473J. Lemma
			473K. Poincaré's inequality for balls
			473L. Corollary
			473M. The case r = 1
			473X. Basic exercises
			473Y. Further exercises
			473 Notes and comments
		474. The distributional perimeter
			474A. Notation
			474B. The divergence of a vector field
			474C. Invariance under isometries
			474D. Locally finite perimeter
			474E. Theorem
			474F. Definitions
			474G. The reduced boundary
			474H. Invariance under isometries
			474I. Half-spaces
			474J. Lemma
			474K. Lemma
			474L .Two isoperimetric inequalities
			474M. Lemma
			474N. Lemma
			474O. Definition
			474P. Lemma
			474Q. Lemma
			474R. Theorem
			474S. Corollary
			474T. The Compactness Theorem
			474X. Basic exercises
			474Y. Further exercises
			474 Notes and comments
		475. The essential boundary
			475A. Notation
			475B. The essential boundary
			475C. Lemma
			475D. Lemma
			475E. Lemma
			475F. Lemma
			475G. Theorem
			475H. Proposition
			475I. Lemma
			475J. Lemma
			475K. Lemma
			475L. Theorem
			475M. Corollary
			475N. The Divergence Theorem
			475O.
			475P. Lemma
			475Q. Theorem
			475R. Convex sets in R^r
			475S. Corollary: Cauchy's Perimeter Theorem
			475T. Corollary: the Convex Isoperimetric Theorem
			475X. Basic exercises
			475Y. Further exercises
			475 Notes and comments
		476. Concentration of measure
			476A. Proposition
			476B. Lemma
			476C. Proposition
			476D. Concentration by partial reflection
			476E. Lemma
			476F. Theorem
			476G. Theorem
			476H. The Isoperimetric Theorem
			476I. Spheres in inner product spaces
			476J. Lemma
			476K.
			476L. Corollary
			476X. Basic exercises
			476Y. Further exercises
			476 Notes and comments
		477. Brownian motion
			477A. Brownian motion
			477B.
			*477C.
			477D. Multidimensional Brownian motion
			477E. Invariant transformations of Wiener measure
			477F. Proposition
			477G. The strong Markov property
			477H. Some families of σ-algebras
			477I. Hitting times
			477J.
			477K. Typical Brownian paths
			477L. Theorem
			477X. Basic exercises
			477Y. Further exercises
			477 Notes and comments
		478. Harmonic functions
			478A. Notation
			478B. Harmonic and superharmonic functions
			478C. Elementary facts
			478D. Maximal principle
			478E. Theorem
			478F. Basic examples
			478G.
			478H. Corollary
			478I.
			478J. Convolutions and smoothing
			478K. Dynkin's formula
			478L. Theorem
			478M. Proposition
			478N. Wandering paths
			478O. Theorem
			478P. Harmonic measures
			478Q.
			478R. Theorem
			478S. Corollary
			478T. Corollary
			478U.
			*478V. Theorem
			478X. Basic exercises
			478Y. Further exercises
			478 Notes and comments
		479. Newtonian capacity
			479A. Notation
			479B. Theorem
			479C. Definitions
			479D.
			479E. Theorem
			479F.
			479G.
			479H.
			479I. Proposition
			479J.
			479K. Lemma
			479L.
			479M.
			479N.
			479O. Polar sets
			479P.
			479Q. Hausdorff measure
			479R.
			479S.
			*479T.
			*479U. Theorem
			*479V.
			*479W.
			479X. Basic exercises
			479Y. Further exercises
			479 Notes and comments
	Chapter 48. Gauge integrals
		481. Tagged partitions
			481A. Tagged partitions and Riemann sums
			481B. Notation
			481C. Proposition
			481D. Remarks
			481E. Gauges
			481F. Residual sets
			481G. Subdivisions
			481H. Remarks
			481I.
			481J. The Henstock integral on a bounded interval (HENSTOCK 63)
			481K. The Henstock integral on R
			481L. The symmetric Riemann-complete integral (cf. CARRINGTON 72, chap. 3)
			481M. The McShane integral on an interval (McSHANE 73)
			481N. The McShane integral on a topological space (FREMLIN 95)
			481O. Convex partitions in R^r
			481P. Box products (cf. MULDOWNEY 87, Prop. 1)
			481Q. The approximately continuous Henstock integral (GORDON 94, chap. 16)
			481X. Basic exercises
			481Y. Further exercises
			481 Notes and comments
		482. General theory
			482A. Lemma
			482B. Saks-Henstock Lemma
			482C. Definition
			482D.
			482E. Theorem
			482F. Proposition
			482G. Proposition
			482H. Proposition
			482I. Integrating a derivative
			482J. Definition
			482K. B.Levi's theorem
			482L. Lemma
			482M. Fubini's theorem
			482X. Basic exercises
			482Y. Further exercises
			482 Notes and comments
		483. The Henstock integral
			483A. Definition
			483B.
			483C. Corollary
			483D. Corollary
			483E. Definition
			483F.
			483G. Theorem
			483H. Upper and lower derivates
			483I. Theorem
			483J. Theorem
			483K. Proposition
			483L. Definition
			483M. Proposition
			483N. Proposition
			483O. Definitions
			483P. Elementary results
			483Q. Lemma
			483R. Theorem
			483X. Basic exercises
			483Y. Further exercises
			483 Notes and comments
		484. The Pfeffer integral
			484A. Notation
			484B. Theorem (TAMANINI & GIACOMELLI 89)
			484C. Lemma
			484D. Definitions
			484E. Lemma
			484F. A family of tagged-partition structures
			484G. The Pfeffer integral
			484H.
			484I. Definition
			484J.
			484K. Lemma
			484L. Proposition
			484M. Lemma
			484N. Pfeffer's Divergence Theorem
			484O. Differentiating the indefinite integral
			484P. Lemma
			484Q. Definition
			484R. Lemma
			484S. Theorem
			484X. Basic exercises
			484Y. Further exercises
			484 Notes and comments
	Chapter 49. Further topics
		491. Equidistributed sequences
			491A. The asymptotic density ideal
			491B. Equidistributed sequences
			491C.
			491D.
			491E. Proposition
			491F. Theorem
			491G. Corollary
			491H. Theorem (VEECH 71)
			491I. The quotient PN/Z
			491J. Lemma
			491K. Corollary
			491L. Effectively regular measures
			491M. Examples
			491N. Theorem
			491O. Proposition
			491P. Proposition
			491Q. Corollary
			491R.
			491S. The asymptotic density filter
			491X. Basic exercises
			491Y. Further exercises
			491Z. Problem
			491 Notes and comments
		492. Combinatorial concentration of measure
			492A. Lemma
			492B. Corollary
			492C. Lemma
			492D. Theorem (TALAGRAND 95)
			492E. Corollary
			492F.
			492G. Lemma (MILMAN & SCHECHTMAN 86)
			492H. Theorem (MAUREY 79)
			492I. Corollary
			492X. Basic exercises
			492 Notes and comments
		493. Extremely amenable groups
			493A. Definition
			493B. Proposition
			493C. Theorem
			493D.
			493E. Theorem (PESTOV 02)
			493F.
			493G. Theorem
			493H.
			493X. Basic exercises
			493Y. Further exercises
			493 Notes and comments
		494. Groups of measure-preserving automorphisms
			494A. Definitions (HALMOS 56)
			494B. Proposition
			494C. Proposition
			494D. Lemma
			494E. Theorem (HALMOS 44, ROKHLIN 48)
			494F.
			494G. Proposition
			494H. Proposition
			494I.
			494J. Lemma
			494K. Lemma
			494L. Theorem
			494M. Lemma
			494N. Lemma
			494O. Theorem (KITTRELL & TSANKOV 09)
			494P. Remark
			494Q.
			494R.
			494X. Basic exercises
			494Y. Further exercises
			494Z. Problems
			494 Notes and comments
		495. Poisson point processes
			495A. Poisson distributions
			495B. Theorem
			495C. Lemma
			495D. Theorem
			495E. Definition
			495F. Proposition
			495G. Proposition
			495H. Lemma
			495I. Theorem
			495J. Proposition
			495K. Proposition
			495L.
			495M.
			495N.
			495O. Proposition
			495P.
			495X. Basic exercises
			495Y. Further exercises
			495 Notes and comments
		496. Maharam submeasures
			496A. Definitions
			496B. Basic facts
			496C. Radon submeasures
			496D. Proposition
			496E. Theorem
			496F. Theorem
			496G. Theorem
			496H. Theorem
			496I. Theorem
			496J. Theorem
			496K. Proposition
			496L. Free products of Maharam algebras
			496M. Representing products of Maharam algebras
			496X. Basic exercises
			496Y. Further exercises
			496 Notes and comments
		497. Tao's proof of Szemerédi's theorem
			497A. Definitions
			497B. Lemma
			497C. Lemma
			497D. Lemma
			497E. Theorem (TAO 07)
			497F. Invariant measures on P([I]^{<ω})
			497G. Theorem (TAO 07)
			497H.
			497I. Definition
			497J. Theorem (NAGLE RÖDL & SCHACHT 06)
			497K. Corollary: the Hypergraph Removal Lemma
			497L. Corollary: Szemerédi's Theorem (SZEMERÉDI 75)
			497M.
			497N. Theorem (FURSTENBURG 81)
			497X. Basic exercises
			497Y. Further exercises
			497 Notes and comments
		498. Cubes in product spaces
			498A. Proposition
			498B. Proposition (see BRODSKIĬ 49, EGGLESTON 54)
			498C. Proposition (see CIESIELSKI & PAWLIKOWSKI 03)
			498X. Basic exercises
			498Y. Further exercises
			498 Notes and comments
	Appendix to Volume 4 - Useful Facts
		4A1. Set theory
			4A1A. Cardinals again
			4A1B. Closed cofinal sets
			4A1C. Stationary sets
			4A1D. Δ-systems
			4A1E. Free sets
			4A1F. Selecting subsequences
			4A1G. Ramsey's theorem
			4A1H. The Marriage Lemma again
			4A1I. Filters
			4A1J. Lemma
			4A1K. Theorem
			4A1L. Theorem
			4A1M. Ostaszewski's
			4A1N. Lemma
			4A1O. The size of σ-algebras
			4A1P. An incidental fact
		4A2. General topology
			4A2A. Definitions
			4A2B. Elementary facts about general topological spaces
			4A2C. G_δ, F_σ, zero and cozero sets
			4A2D. Weight
			4A2E. The countable chain condition
			4A2F. Separation axioms
			4A2G. Compact and locally compact spaces
			4A2H. Lindelöf spaces
			4A2I. Stone-Čech compactifications
			4A2J. Uniform spaces
			4A2K. First-countable, sequential and countably tight spaces
			4A2L. (Pseudo-)metrizable spaces
			4A2M. Complete metric spaces
			4A2N. Countable networks
			4A2O. Second-countable spaces
			4A2P. Separable metrizable spaces
			4A2Q. Polish spaces
			4A2R. Order topologies
			4A2S. Order topologies on ordinals
			4A2T. Topologies on spaces of subsets
			4A2U. Old friends
		4A3. Topological σ-algebras
			4A3A. Borel sets
			4A3B. (Σ, T)-measurable functions
			4A3C. Elementary facts
			4A3D. Hereditarily Lindelöf spaces
			4A3E. Applications
			4A3F. Spaces with countable networks
			4A3G. Second-countable spaces
			4A3H. Borel sets in Polish spaces
			4A3I. Corollary
			4A3J. Borel sets in ω_1
			4A3K. Baire sets
			4A3L. Lemma
			4A3M. Product spaces
			4A3N. Products of separable metrizable spaces
			4A3O. Compact spaces
			4A3P. Proposition
			4A3Q. Baire property
			4A3R. Proposition
			*4A3S.
			4A3T. Cylindrical σ-algebras
			4A3U. Proposition
			4A3V. Proposition
			4A3W. Càdlàg functions
			4A3X. Basic exercises
			4A3Y. Further exercises
			4A3 Notes and comments
		4A4. Locally convex spaces
			4A4A. Linear spaces
			4A4B. Linear topological spaces
			4A4C. Locally convex spaces
			4A4D. Hahn-Banach theorem
			4A4E. The Hahn-Banach theorem in locally convex spaces
			4A4F. The Mackey topology
			4A4G. Extreme points
			4A4H. Proposition
			4A4I. Normed spaces
			4A4J. Inner product spaces
			4A4K. Hilbert spaces
			4A4L. Compact operators
			4A4M. Self-adjoint compact operators
			4A4N. Max-flow Min-cut Theorem (FORD & FULKERSON 56)
		4A5. Topological groups
			4A5A. Notation
			4A5B. Group actions
			4A5C. Examples
			4A5D. Definitions
			4A5E. Elementary facts
			4A5F. Proposition
			4A5G. Proposition
			4A5H. The uniformities of a topological group
			4A5I. Definitions
			4A5J. Quotients under group actions, and quotient groups
			4A5K. Proposition
			4A5L. Theorem
			4A5M. Proposition
			4A5N. Theorem
			4A5O. Proposition
			4A5P. Lemma
			4A5Q. Metrizable groups
			4A5R. Corollary
			4A5S. Lemma
			*4A5T.
		4A6. Banach algebras
			4A6A. Definition
			4A6B. Stone-Weierstrass Theorem: fourth form
			4A6C. Proposition
			4A6D. Proposition
			4A6E. Proposition
			4A6F. Proposition
			4A6G. Definition
			4A6H. Theorem
			4A6I. Theorem
			4A6J. Theorem
			4A6K. Corollary
			4A6L. Exponentiation
			4A6M. Lemma
			4A6N. Lemma
			4A6O. Proposition
		4A7. `Later editions only'
	Concordance to chapters 46-49
	References for Volume 4
	Index to volumes 1-4
		Principal topics and results
		General index
Measure Theory 5-1_Set-theoretic Measure Theory(2015,329p)D.H.Fremlin_9780953812950
	Contents
		General introduction
		Introduction to Volume 5
		Note on second printing
	Chapter 51. Cardinal functions
		511. Definitions
			511A. Pre-ordered sets
			511B. Definitions
			511C. On the symbol ∞
			511D. Definitions
			511E. Precalibers
			511F. Definitions
			511G. Definition
			511H. Elementary facts: pre-ordered sets
			511I. Elementary facts: Boolean algebras
			511J. Elementary facts: ideals of sets
			511X. Basic exercises
			511Y. Further exercises
			511 Notes and comments
		512. Galois-Tukey connections
			512A. Definitions
			512B. Definitions
			512C.
			512D. Theorem
			512E. Examples
			512F.
			512G. Proposition
			512H. Simple products
			512I. Sequential compositions
			512J. Proposition
			512K.
			512X. Basic exercises
			512 Notes and comments
		513. Partially ordered sets
			513A.
			513B. Theorem
			513C. Cofinalities of cardinal functions
			513D.
			513E. Theorem
			513F. Theorem (TUKEY 40)
			513G.
			513H. Definition
			513I. Proposition
			*513J. Cofinalities of products
			*513K.
			*513L. Proposition
			*513M. Proposition
			*513N. Lemma
			*513O. Theorem (SOLECKI & TODORČEVIC 04)
			513P.
			513X. Basic exercises
			513Y. Further exercises
			513 Notes and comments
		514. Boolean algebras
			514A.
			514B. Stone spaces
			514C.
			514D. Theorem
			514E. Subalgebras, homomorphic images, products
			514F.
			514G. Order-preserving functions of Boolean algebras
			514H. Regular open algebras
			514I. Category algebras
			514J.
			514K.
			514L. The regular open algebra of a pre-ordered set
			514M.
			514N. Proposition
			514O.
			514P. Corollary
			514Q. Proposition
			514R. Corollary
			514S. Proposition
			514T. Finite-support products
			514U. Proposition
			514X. Basic exercises
			514Y. Further exercises
			514. Notes and comments
		515. The Balcar-Franĕk theorem
			515A. Definition
			515B. Lemma
			515C. Proposition
			515D. Lemma
			515E. Lemma (BALCAR & VOITÁŠ 77)
			515F. Lemma
			515G. Lemma
			515H. The Balcar-Franĕk theorem (BALCAR & FRANĔK 82)
			515I. Corollary
			515J. Corollary
			515K.
			515L. Theorem (KOPPELBERG 75)
			515M. Corollary
			515N.
			515X. Basic exercises
			515Y. Further exercises
			515 Notes and comments
		516. Precalibers
			516A. Definition
			516B. Elementary remarks
			516C. Theorem
			516D. Corollary
			516E. Remark
			516F.
			516G. Corollary
			516H. Corollary
			516I. Corollary
			516J.
			516K.
			516L. Corollary
			516M.
			516N. Corollary
			516O.
			516P. Corollary
			516Q.
			516R.
			516S.
			516T.
			516U.
			516X. Basic exercises
			516 Notes and comments
		517. Martin numbers
			517A. Proposition
			517B. Lemma
			517C. Lemma
			517D. Proposition
			517E. Corollary
			517F. Proposition
			517G. Corollary
			517H. Proposition
			517I. Proposition
			517J. Proposition
			517K. Corollary
			517L.
			517M.
			517N. Corollary
			517O. Martin cardinals
			517P.
			517Q. Lemma
			517R. Proposition
			517S. Proposition
			517X. Basic exercises
			517Y. Further exercises
			517 Notes and comments
		518. Freese-Nation numbers
			518A. Proposition (FUCHINO KOPPELBERG & SHELAH 96)
			518B. Proposition
			518C. Corollary
			518D. Corollary
			518E.
			518F. Lemma
			518G. Lemma (FUCHINO KOPPELBERG & SHELAH 96)
			518H. Lemma
			518I. Theorem (FUCHINO & SOUKUP 97)
			518J. Lemma
			518K. Theorem (FUCHINO GESCHKE SHELAH & SOUKUP 01)
			518L.
			518M. Theorem
			518N. Definition
			518O. Lemma
			518P. Lemma (GESCHKE 02)
			518Q. Corollary
			518R. Lemma
			518S. Theorem (GESCHKE 02)
			518X. Basic exercises
			518Y. Further exercises
			518 Notes and comments
	Chapter 52. Cardinal functions of measure theory
		521. Basic theory
			521A. Proposition
			521B. Proposition
			521C.
			521D. Proposition
			521E.
			521F. Proposition
			521G. Proposition
			521H. Proposition
			521I. Corollary
			521J.
			521K.
			521L. Proposition
			521M. Proposition
			521N. Proposition
			521O. Proposition
			521P. Proposition
			521Q. Free products
			521R. Proposition
			521S. Proposition
			521T.
			521X. Basic exercises
			521Y. Further exercises
			521 Notes and comments
		522. Cichoń's diagram
			522A. Notation
			522B. Cichoń's diagram
			522C. Lemma
			522D. Proposition
			522E. Proposition
			522F. Proposition
			522G. Proposition (ROTHBERGER 38A)
			522H. Proposition
			522I. Proposition
			522J. Theorem (see TRUSS 77 and MILLER 81)
			522K. Localization
			*522L. Lemma
			522M. Proposition
			522N. Lemma
			522O. Proposition
			522P. Corollary
			522Q. Theorem (BARTOSZYŃSKI 84, RAISONNIER & STERN 85)
			522R. The exactness of Cichoń's diagram
			522S. The cardinal cov M
			522T. Martin numbers
			*522U. FN(PN)
			522V. Cofinalities
			522W. Other spaces
			522X. Basic exercises
			522Y. Further exercises
			522 Notes and comments
		523. The measure of {0, 1}^{I}
			523A. Notation
			523B. The basic diagram
			523C.
			523D.
			523E. Additivities
			523F. Covering numbers
			523G.
			523H. Uniformities
			523I. Theorem
			523J. Corollary (KRASZEWSKI 01)
			523K. Corollary (BURKE N05)
			523L.
			523M. Shrinking numbers
			523N. Cofinalities
			523O. Cofinalities of the cardinals
			523P. The generalized continuum hypothesis
			523X. Basic exercises
			523Y. Further exercises
			523Z. Problem
			523 Notes and comments
		524. Radon measures
			524A. Notation
			524B. Proposition
			524C. Lemma
			524D. Proposition
			524E. Proposition
			524F. Lemma
			524G. Proposition
			524H. Corollary
			524I. Corollary
			524J. Theorem
			524K. Corollary
			524L.
			524M. Theorem
			524N. Corollary
			524O. Freese-Nation numbers
			524P. The Maharam classification
			*524Q.
			524R.
			524S.
			524T. Corollary
			524X. Basic exercises
			524Y. Further exercises
			524Z. Problems
			524 Notes and comments
		525. Precalibers
			525A. Notation
			525B. Proposition
			525C. Theorem
			525D. Proposition
			525E. Proposition
			525F. Proposition
			525G.
			525H. The structure of B_I
			525I. Theorem
			525J. Corollary
			525K. Proposition
			525L.
			525M. Proposition
			525N. Proposition (ARGYROS & TSARPALIAS 82)
			525O.
			*525P.
			525Q.
			525R. Lemma
			525S. Theorem (FREMLIN 88)
			525T. Corollary (ARGYROS & KALAMIDAS 82)
			525X. Basic exercises
			525Z. Problem
			525 Notes and comments
		526. Asymptotic density zero
			526A. Proposition
			526B. Proposition (FREMLIN 91)
			526C.
			526D. Lemma
			526E. Lemma
			526F. Theorem
			526G. Corollary
			526H.
			526I.
			526J. Proposition
			526K. Proposition
			526L. Proposition (MÁTRAI P09)
			526M.
			526X. Basic exercises
			526Y. Further exercises
			526 Notes and comments
		527. Skew products of ideals
			527A. Notation
			527B. Skew products of ideals
			527C.
			527D.
			527E. Corollary
			527F.
			527G. Theorem
			527H. Corollary
			527I.
			527J. Theorem (see FREMLIN 91)
			527K. Corollary
			527L.
			527M.
			527N. Lemma
			527O. Theorem
			527X. Basic exercises
			527Y. Further exercises
			527 Notes and comments
		528. Amoeba algebras
			528A. Amoeba algebras
			528B.
			528C. Proposition
			528D. Proposition
			528E. Lemma
			528F. Proposition
			528G. Proposition
			528H. Proposition
			528I. Definition
			528J. Proposition
			528K. Theorem (TRUSS 88)
			528L.
			528M. Lemma
			528N. Theorem (BRENDLE 00, 2.3.10; JUDAH & REPICKÝ 95)
			528O. Corollary
			528P. Proposition
			528Q. Proposition
			528R. Theorem
			528S.
			528T. Lemma
			528U. Lemma
			528V. Theorem
			528X. Basic exercises
			528Y. Further exercises
			528Z. Problems
			528 Notes and comments
		529. Further partially ordered sets of measure theory
			529A. Notation
			529B. Proposition
			529C. Theorem (FREMLIN 91)
			529D. Theorem (FREMLIN 91)
			529E. Proposition
			529F. Corollary (BRENDLE 00, 2.3.10; BRENDLE 06, Theorem 1)
			529G. Reaping numbers (following BRENDLE 00)
			529H. Proposition (BRENDLE 00, 2.7; BRENDLE 06, Theorem 5)
			529X. Basic exercises
			529Y. Further exercises
			529 Notes and comments
	Chapter 53. Topologies and measures III
		531. Maharam types of Radon measures
			531A. Proposition
			531B.
			531C. Lemma
			531D. Definition
			531E. Proposition
			531F. Proposition
			531G. Proposition
			531H. Remarks
			531I. Notation
			531J. Lemma
			531K. Lemma
			531L. Theorem
			531M. Proposition (PLEBANEK 97)
			531N.
			531O.
			531P.
			531Q.
			531R.
			531S.
			531T. Theorem (FREMLIN 97)
			531X. Basic exercises
			531Y. Further exercises
			531Z. Problems
			531 Notes and comments
		532. Completion regular measures on {0, 1}^{I}
			532A. Definition
			532B. Proposition
			532C. Remarks
			532D. Theorem (FREMLIN & GREKAS 95)
			532E. Corollary
			532F. Corollary
			532G. Proposition
			532H. Lemma
			532I.
			532J. Corollary
			532K. Corollary
			532L. Corollary
			532M.
			532N.
			532O. Proposition
			532P. Proposition
			532Q. Proposition
			532R.
			532S. Proposition
			532X. Basic exercises
			532Y. Further exercises
			532Z. Problems
			532 Notes and comments
		533. Special topics
			533A. Lemma
			533B. Corollary
			533C. Proposition
			533D. Proposition
			533E. Corollary
			533F. Definition
			533G. Lemma
			533H. Theorem
			533I.
			533J. Theorem (see FREMLIN 77)
			533X. Basic exercises
			533Y. Further exercises
			533Z. Problem
			533 Notes and comments
		534. Hausdorff measures and strong measure zero
			534A.
			534B. Hausdorff measures
			534C. Strong measure zero
			534D. Proposition
			534E. Rothberger's property
			534F. Proposition
			534G. Corollary
			534H. Theorem
			534I. Proposition
			534J. Proposition (FREMLIN 91)
			534K. Corollary
			534L. Smz-equivalence
			534M. Lemma
			534N. Proposition
			534O. Large sets with strong measure zero
			534P.
			534X. Basic exercises
			534Y. Further exercises
			534Z. Problems
			534 Notes and comments
		535. Liftings
			535A. Notation
			535B. Proposition
			535C. Proposition
			535D.
			535E. Proposition
			535F.
			535G. Corollary (see NEUMANN 31)
			535H.
			535I. Corollary (see MOKOBODZKI 75)
			535J.
			535K. Lemma
			535L. Lemma
			535M. Lemma
			535N. Theorem
			535O. Linear liftings
			535P.
			535Q. Proposition
			535R. Proposition
			535X. Basic exercises
			535Y. Further exercises
			535Z. Problems
			535 Notes and comments
		536. Alexandra Bellow's problem
			536A. The problem
			536B. Known cases
			536C. Proposition (see TALAGRAND 84, 9-3-3.)
			536D. Theorem
			536X. Basic exercises
			536Y. Further exercises
			536 Notes and comments
		537. Sierpiński sets, shrinking numbers and strong Fubini theorems
			537A. Definitions
			537B. Proposition
			537C. Entangled sets
			537D. Lemma
			537E. Lemma
			537F. Corollary
			537G. Theorem (TODORČEVIĆ 85)
			537H. Scalarly measurable functions
			537I. Proposition
			537J. Corollary
			537K.
			537L. Corollary
			537M.
			537N.
			537O. Corollary
			537P. Corollary
			537Q.
			537R. Lemma
			537S. Proposition
			537X. Basic exercises
			537Z. Problems
			537 Notes and comments
		538. Filters and limits
			538A. Filters
			538B.
			538C. Lemma
			538D. Finite products of filters
			538E.
			538F. Ramsey filters
			538G. Measure-centering filters
			538H. Proposition
			538I. Theorem
			538J. Proposition
			538K.
			538L. Theorem
			538M. Benedikt's theorem (BENEDIKT 98)
			538N. Measure-converging filters
			538O. The Fatou property
			538P. Theorem
			538Q. Definition
			538R. Proposition
			538S. Theorem
			538X. Basic exercises
			538Y. Further exercises
			538Z. Problem
			538 Notes and comments
		539. Maharam submeasures
			539A. The story so far
			539B. Proposition
			539C. Theorem
			539D. Corollary
			539E. Proposition (VELIČKOVIĆ 05, BALCAR JECH & PAZÁK 05)
			539F. Definition
			539G. Proposition
			539H. Corollary
			539I. Corollary
			539J. Theorem
			539K.
			539L.
			539M. Lemma
			539N. Theorem (BALCAR JECH & PAZÁK 05, VELIČKOVIĆ 05)
			539O. Corollary
			539P.
			539Q. Reflection principles
			539R. Exhaustivity rank
			539S. Elementary facts
			539T. The rank of a Maharam algebra
			539U. Theorem
			539X. Basic exercises
			539Y. Further exercises
			539Z. Problems
			539 Notes and comments
	Concordance to chapters 51-53
Measure Theory 5-2_Set-theoretic Measure Theory(2015,411p)D.H.Fremlin_9780953812967
	Contents
	Chapter 54. Real-valued-measurable cardinals
		541. Saturated ideals
			541A. Definition
			541B. Proposition
			541C. Proposition
			541D. Lemma
			541E. Corollary
			541F. Lemma
			541G. Definition
			541H. Proposition
			541I. Lemma
			541J. Theorem (SOLOVAY 71)
			541K. Lemma
			541L. Theorem
			541M. Definition
			541N. Theorem
			541O. Lemma
			541P. Theorem (TARSKI 45, SOLOVAY 71)
			541Q. Theorem
			541R. Corollary
			541S. Lemma
			541X. Basic exercises
			541Y. Further exercises
			541 Notes and comments
		542. Quasi-measurable cardinals
			542A. Definition
			542B. Proposition
			542C. Proposition
			542D. Proposition
			542E. Theorem (GITIK & SHELAH 93)
			542F. Corollary
			542G. Corollary
			542H. Lemma
			542I. Theorem (SHELAH 96)
			542J. Corollary
			542X. Basic exercises
			542Y. Further exercises
			542 Notes and comments
		543. The Gitik-Shelah theorem
			543A. Definitions
			543B.
			543C. Theorem (see KUNEN N70)
			543D. Corollary
			543E. The Gitik-Shelah theorem (GITIK & SHELAH 89, GITIK & SHELAH 93)
			543F. Theorem
			543G. Corollary
			543H. Corollary
			543I. Corollary
			543J. Proposition
			543K. Proposition
			543L. Proposition
			543X. Basic exercises
			543Y. Further exercises
			543Z. Problems
			543 Notes and comments
		544. Measure theory with an atomlessly-measurable cardinal
			544A. Notation
			544B. Proposition
			544C. Theorem (KUNEN N70)
			544D. Corollary
			544E. Theorem (KUNEN N70)
			544F. Theorem (KUNEN N70)
			544G. Proposition
			544H. Corollary
			544I.
			544J. Proposition (ZAKRZEWSKI 92)
			544K. Proposition
			544L. Corollary
			544M. Theorem
			544N. Cichoń's diagram and other cardinals
			544X. Basic exercises
			544Y. Further exercises
			544Z. Problems
			544 Notes and comments
		545. PMEA and NMA
			545A. Theorem
			545B. Definition
			545C. Proposition
			545D. Definition
			545E. Proposition
			545F. Proposition
			545G. Corollary
			545X. Basic exercises
			545Y. Further exercises
			545 Notes and comments
		546 Power set σ-quotient algebras
			546A.
			546B. Lemma
			546C.
			546D.
			546E.
			546F. Corollary
			546G. The Gitik-Shelah theorem for Cohen algebras
			546H.
			546I. Corollary
			546J.
			546K. Lemma
			546L.
			546M. Theorem
			546N. Lemma
			546O. Lemma
			546P. Theorem
			546Q. Corollary
			546X. Basic exercises
			546Y. Further exercises
			546Z. Problems
			546 Notes and comments
		547. Disjoint refinements of sequences of sets
			547A. Lemma
			547B. Lemma
			547C. Lemma
			547D. Lemma
			547E. Lemma
			547F. Theorem
			547G. Corollary
			547H.
			547I. Proposition
			547J. Corollary
			547X. Basic exercises
			547Z. Problems
			547 Notes and comments
	Chapter 55. Possible worlds
		551. Forcing with quotient algebras
			551A. Definition
			551B. Definition
			551C. Definition
			551D. Definition
			551E. Proposition
			551F. Proposition
			551G.
			551H. Examples
			551I. Theorem
			551J. Corollary
			551K.
			551L. Remark
			551M.
			551N. Proposition
			551O. Measure algebras
			551P. Theorem
			551Q. Iterated forcing
			551R. Extending filters
			551X. Basic exercises
			551Y. Further exercises
			551 Notes and comments
		552. Random reals I
			552A. Notation
			552B. Theorem
			552C. Theorem
			552D. Lemma
			552E. Theorem
			552F. Theorem
			552G. Theorem
			552H. Theorem
			552I. Theorem
			552J. Theorem
			552K. Lemma
			552L. Lemma
			552M. Proposition
			552N. Theorem (CARLSON 84)
			552O. Proposition
			552P. Theorem
			552X. Basic exercises
			552Y. Further exercises
			552 Notes and comments
		553. Random reals II
			553A. Notation
			553B. Lemma
			553C. Proposition
			553D. Remark
			553E. Proposition
			553F. Corollary
			553G. Lemma
			553H. Theorem
			553I. Lemma
			553J. Theorem
			553K.
			553L. Lemma
			553M. Proposition (LAVER 87)
			553N. Proposition
			553O.
			553X. Basic exercises
			553Y. Further exercises
			553Z. Problem
			553 Notes and comments
		554. Cohen reals
			554A. Notation
			554B. Theorem
			554C. Definition
			554D. Proposition
			554E. Theorem
			554F. Corollary
			554G. Theorem
			554H. Corollary
			554I. Theorem (CARLSON FRANKIEWICZ & ZBIERSKI 94)
			554X. Basic exercises
			554Y. Further exercises
			554 Notes and comments
		555. Solovay's construction of real-valued-measurable cardinals
			555A. Notation
			555B. Theorem
			555C. Theorem
			555D. Corollary (SOLOVAY 71)
			555E. Theorem
			555F. Proposition
			555G. Cohen forcing
			555H. Corollary
			555I.
			555J. Lemma
			555K. Główczyński's example (GŁÓWCZYŃSKI 91, BALCAR JECH & PAZÁK 05, GŁÓWCZYŃSKI 08)
			555L. Supercompact cardinals and the normal measure axiom
			555M. Proposition
			555N. Theorem (PRIKRY 75, FLEISSNER 91)
			555O.
			555X. Basic exercises
			555Y. Further exercises
			555Z. Problems
			555 Notes and comments
		556. Forcing with Boolean subalgebras
			556A. Forcing with Boolean subalgebras
			556B. Theorem
			556C. Theorem
			556D. Regularly embedded subalgebras
			556E. Proposition
			556F. Quotient forcing
			556G. Proposition
			556H. L^0(\mathfrak{A})
			556I. Proposition
			556J. Theorem
			556K. Theorem
			556L. Relatively independent subalgebras
			556M. Laws of large numbers
			556N. Dye's theorem
			556O.
			556P. Kawada's theorem
			556Q.
			556R. Proposition
			556S. Theorem (FARAH 06)
			556X. Basic exercises
			556Y. Further exercises
			556 Notes and comments
	Chapter 56. Choice and determinacy
		561. Analysis without choice
			561A. Set theory without choice
			561B. Real analysis without choice
			561C.
			561D. Tychonoff's theorem
			561E. Baire's theorem
			561F. Stone's Theorem
			561G. Haar measure
			561H. Kakutani's theorem
			561I. Hilbert spaces
			561X. Basic exercises
			561Y. Further exercises
			561 Notes and comments
		562. Borel codes
			562A. Trees
			562B. Coding sets with trees
			562C.
			562D. Proposition
			562E. Proposition
			*562F.
			562G. Resolvable sets
			562H. Proposition
			562I. Theorem
			562J. Codable families of sets
			562K. Proposition
			562L. Codable Borel functions
			562M. Theorem
			562N. Proposition
			562O. Remarks
			562P. Codable Borel equivalence
			562Q. Resolvable functions
			562R. Theorem
			562S. Codable families of codable functions
			562T. Codable Baire sets
			562U. Proposition
			562V.
			562X. Basic exercises
			562Y. Further exercises
			562 Notes and comments
		563. Borel measures without choice
			563A. Definitions
			563B. Proposition
			563C. Corollary
			563D.
			563E. Lemma
			563F. Proposition
			563G. Proposition
			563H. Theorem
			563I. Theorem
			563J. Baire-coded measures
			563K. Proposition
			563L. Proposition
			563M. Measure algebras
			563N. Theorem
			563X. Basic exercises
			563Z. Problem
			563 Notes and comments
		564. Integration without choice
			564A. Definitions
			564B. Lemma
			564C. Definition
			564D. Lemma
			564E. Theorem
			564F.
			564G. Integration over subsets
			564H. Theorem
			564I. Riesz Representation Theorem
			564J. The space L^1
			564K.
			564L. Radon-Nikodým theorem
			564M. Inverse-measure-preserving functions
			564N. Product measures
			564O. Theorem
			564X. Basic exercises
			564Y. Further exercises
			564 Notes and comments
		565. Lebesgue measure without choice
			565A. Definitions
			565B. Proposition
			565C. Lemma
			565D. Definition
			565E. Proposition
			565F. Vitali's Theorem
			565G. Proposition
			565H. Corollary
			565I. Lemma
			565J. Lemma
			565K. Theorem
			565L. Lemma
			565M. Theorem
			565N. Hausdorff measures
			565O. Theorem
			565X. Basic exercises
			565Y. Further exercises
			565 Notes and comments
		566. Countable choice
			566A.
			566B. Volume 1
			566C. Volume 2
			566D. Exhaustion
			566E.
			566F. Atomless algebras
			566G. Vitali's theorem
			566H. Bounded additive functionals
			566I. Infinite products
			566J.
			566K. Volume 3
			566L. The Loomis-Sikorski theorem
			566M. Measure algebras
			566N. Characterizing the usual measure on {0, 1}^N
			566O. Boolean values
			566P. Weak compactness
			566Q. Theorem [AC(ω)]
			566R. Automorphisms of measurable algebras
			566S. Volume 4
			566T.
			566U. Dependent choice
			566X. Basic exercises
			566Y. Further exercises
			566Z. Problem
			566 Notes and comments
		567. Determinacy
			567A. Infinite games
			567B. Theorem
			567C. The axiom of determinacy
			567D. Theorem (MYCIELSKI 64)
			567E. Consequences of AC(R; ω)
			567F. Lemma (see MYCIELSKI & ŚWIERCZKOWSKI 64) [AC(R; ω)]
			567G. Theorem [AD]
			567H. Theorem
			567I. Proposition [AC(R; ω)]
			567J. Proposition [AD]
			567K. Theorem [AD+AC(ω)]
			567L. Theorem (R.M.Solovay) [AD]
			567M. Theorem (MOSCHOVAKIS 70) [AD]
			567N. Theorem (MARTIN 70) [AC]
			567O. Corollary [AC]
			567X. Basic exercises
			567Y. Further exercises
			567 Notes and comments
	Appendix to Volume 5 - Useful Facts
		5A1. Set theory
			5A1A. Order types
			5A1B. Ordinal arithmetic
			5A1C. Well-founded sets
			5A1D. Trees
			5A1E. Cardinal arithmetic
			5A1F. Three fairly simple facts
			5A1G. Partition calculus - The Erdös-Rado theorem
			5A1H. Δ-systems and free sets
			5A1I. Remarks
			5A1J. Lemma
			5A1K. Lemma
			5A1L. Definitions
			5A1M. Lemma
			5A1N. Almost-square-sequences
			5A1O. Corollary
		5A2. Pcf theory
			5A2A. Reduced products
			5A2B. Theorem
			5A2C. Theorem
			5A2D. Definitions
			5A2E. Lemma
			5A2F. Lemma
			5A2G. Theorem
			5A2H. Lemma
			5A2I. Lemma
		5A3. Forcing
			5A3A. Forcing notions
			5A3B. Forcing languages
			5A3C. The Forcing Relation (KUNEN 80, VII.3.3)
			5A3D. The Forcing Theorem
			5A3E. More notation
			5A3F. Boolean truth values
			5A3G. Concerning š
			5A3H. Names for functions
			5A3I. Regular open algebras
			5A3J.
			5A3K. Lemma
			5A3L. Real numbers in forcing languages
			5A3M. Forcing with Boolean algebras
			5A3N. Ordinals and cardinals
			5A3O. Iterated forcing (KUNEN 80, VIII.5.2)
			5A3P. Martin's axiom
			5A3Q. Countably closed forcings
			5A3 Notes and comments
		5A4. General topology
			5A4A. Definitions
			5A4B. Proposition
			5A4C. Compactness
			5A4D. Vietoris topologies
			5A4E. Category and the Baire property
			5A4F. Normal and paracompact spaces
			5A4G. Baire σ-algebras
			5A4H. Proposition
			5A4I. Old friends
		5A5. Real analysis
			5A5A. Entire functions
		5A6. Special axioms
			5A6A. The generalized continuum hypothesis
			5A6B. L, 0^# and Jensen's Covering Lemma
			5A6C. Theorem
			5A6D. Square principles
			5A6E. Lemma
			5A6F. Chang's transfer principle
			5A6G. Todorčević's p-ideal dichotomy
			*5A6H. Analytic P-ideals
			5A6I. u, g and the filter dichotomy
			*5A6J. Proposition (BLASS & LAFLAMME 89)
	References for Volume 5
	Index to volumes 1-5
		Principal topics and results
		General index
			A
			B
			C
			D
			E
			F
			G
			H
			I
			J・K
			L
			M
			N
			O
			P
			Q・R
			S
			T
			U
			V・W
			Z
		Subject Index
			A
			B・C
			D・E・F
			G・H・I・J・K・L
			M
			N・O・P・Q・R
			S・T・U・V・W・Z
			α・β・γ・δ・θ・λ・μ・ν・π・σ
			τ・υ・φ・χ・ψ・ω
			math symbols