Ulam’s Conjecture on Invariance of Measure in the Hilbert Cube

Preface
Contents
1 Topology
	1.1 Basic Concepts for Metric Spaces
	1.2 Compactness for Metric Spaces
	1.3 Compact Topological Spaces
	1.4 Product of Topological Spaces
	1.5 Completeness
	1.6 Separation Properties
	1.7 Open and Closed Subsets
2 Hilbert Spaces
	2.1 Vector Spaces
	2.2 Basis of Vector Space
	2.3 Normed Spaces
	2.4 Banach Spaces
	2.5 Inner Product Spaces
	2.6 Hilbert Spaces
	2.7 Orthogonal Complements
	2.8 Separable Hilbert Spaces
3 Measure Theory
	3.1 Outer Measures
	3.2 Measures in Abstract Spaces
	3.3 Measures in Metric Spaces
	3.4 Metric Outer Measures
	3.5 Lebesgue Measures
	3.6 Hausdorff Measures
4 Extension of Isometries
	4.1 Basic Concepts and Remarks
	4.2 First-Order Generalized Span
	4.3 First-Order Extension of Isometries
	4.4 Second-Order Generalized Span
	4.5 Basic Cylinders and Basic Intervals
	4.6 Second-Order Extension of Isometries
	4.7 Extension of Isometries to the Entire Space
5 History of Ulam\'s Conjecture
	5.1 Historical Background
	5.2 Basic Definitions
	5.3 Mycielski\'s Partial Solution
	5.4 Fickett\'s Partial Solution
	5.5 Jung and Kim\'s Partial Solution
6 Ulam\'s Conjecture
	6.1 Basic Definitions
	6.2 Cylinders
	6.3 Elementary Volumes
	6.4 Construction of Invariant Measure
	6.5 Efficient Coverings
	6.6 Ulam\'s Conjecture on Invariance of Measure
Bibliography
Index