Set functions

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Preface......Page 4
Contents......Page 6
1. Fundamental relations for sets......Page 9
2. Systems of sets......Page 10
1. Topological spaces......Page 12
2. SoparaWe sets in topological spaces......Page 15
3. Metric spaces......Page 16
1. Additive act functions......Page 19
2. Totally additive set functions......Page 22
3. T xtremes of totally additive act functions......Page 24
4. Positive-, negative-, absolute-function......Page 26
5. Theorems about limits......Page 30
6. Sequences of set functions......Page 31
1. Zero-sets......Page 35
2. Neglecting zero-sets......Page 36
3. Zero-sets in the wider sense.......Page 41
4. Complete fields.  - - - -......Page 42
1. Singular sets......Page 44
2. Regular and singular sot functions......Page 48
3. Continuous and purely discontinuous set functions......Page 51
4. p-Atoms......Page 53
5. Atomlese and atomic set functions......Page 56
6. Intermediate Values.......Page 59
7. k-continuous set functions......Page 62
8. Sequences of 1G-continuous set functions......Page 64
1. Measurable sets......Page 69
2. Outer and inner gyp-measure......Page 72
3. Measure-covers and measure-kernels......Page 76
4. Regular measure functions......Page 80
5. A method for construction of regular measure functions......Page 82
6. Extension of a totally additive set function......Page 88
1. Ordinary measure functions......Page 89
2. Content functions......Page 92
3. Non-w-measurable sets......Page 95
4. Content-like set functions......Page 96
1. Interval functions......Page 98
2. The n-dimensional measure of a point set of R......Page 101
3. Basis of the real numbers......Page 108
4. Onedimensionally non-measurable sets......Page 111
5. A second general method for the construction of content functions......Page 112
6. The k-dimensional measure of a point set of R......Page 114
1. Basic properties ofo-measurable functions......Page 118
2. Relations ofo-measurable functions......Page 121
3. Non-?easurable functions......Page 124
4. c-equivalent functions......Page 126
1. Sequences of measurable functions......Page 127
2. n-convergent sequences......Page 129
3. Asymptotic convergence......Page 134
4. Compact sets offo-measurable functions......Page 143
1. Measurability of continuous functions......Page 147
2.p-continuousfunctions......Page 149
3. Continuity properties of measurable functions......Page 152
1. They-integral of a function......Page 157
2. Other properties of v-integrable functions......Page 166
3. Sununablefunctions......Page 173
4. Characterization of the set functions that are a-integrals......Page 176
5. Upper and lower integral......Page 179
6. Content-like determining function......Page 183
7. Improper integrals......Page 184
1. Lebesgue sums......Page 187
2. Riemann sums......Page 196
3. Distinguished sequences of decomposition......Page 194
4. Darboux sums......Page 198
1. The first mean value theorem......Page 202
2. The second mean value theorem......Page 203
3. The Holder inequality......Page 206
4. The H Olderinequalityforintegrals......Page 210
1. Mean convergence......Page 216
2. Integrable and completely integrable sequences......Page 221
1. Product of monotone set functions.......Page 231
2. Product of non-monotone set functions......Page 242
3. Geometric interpretation of the integral......Page 245
4. Fuhini's theorem......Page 246
5. Iterated integrals......Page 249
1. Derivates ofa set function......Page 254
2. Vitali derivates......Page 255
3. Paving derivates......Page 262
4. The regular derivative......Page 265
5. Covering systems......Page 269
6. Tile derivates......Page 276
1. Density ofa set......Page 282
2. Density ofa set in R......Page 289
3. Approximately continuous functions......Page 294
4. Integration and differentiation......Page 297
5. The limit functions of a derivate......Page 302
6. Transformation of integrals......Page 304
1. Interval functions and associated set functions......Page 305
2. Derivatives of interval functions......Page 310
BIBLIOGRAPHY (LIST of BOOKS QUOTED)......Page 320
LIST OF SYMBOLS AND SIGNS......Page 321
INDEX......Page 325