Preface
Introduction
1 The Notion of Measure 0
1.1 The Riemann Integral
1.2 The Lebesgue Integral
2 Lebesgue Measure on R 13
2.1 Sigma Algebras and Borel Sets
2.2 Definition of a Lebesgue Measure
2.3 There is No Lebesgue Measure on _(P(R)
2.4 Lebesgue Measurable Sets: ML(R) $ _(P(R))
2.5 Calculating Lebesgue Measures
2.6 Approximating Lebesgue Measurable Sets
2.7 Properties of Lebesgue Measure
2.7.1 Regularity
2.7.2 Continuity
2.8 Discussion on B(R) &ML(R)
3 Measurable Functions 55
3.1 Extended Real-Valued Functions
3.2 Equivalent Definitions of Measurability
3.3 Examples of Measurable Functions
3.4 Properties of Measurable Functions
3.4.1 Elementary Function Combinations
3.4.2 Function Sequences
Function Sequence Behaviors
Function Sequence Measurability Properties
3.5 Approximating Lebesgue Measurable Functions
3.6 Distribution Functions of Measurable Functions
4 Littlewood.s Three Principles
4.1 Measurable Sets
4.2 Convergent Sequences of Measurable Functions
4.3 Measurable Functions
5 Borel Measures on R
5.1 Functions Induced by Borel Measures
5.2 Borel Measures from Distribution Functions
5.3 Consistency of Borel Measure Constructions
5.4 Approximating Borel Measurable Sets
5.5 Properties of Borel Measures
5.6 Differentiable F-Length and Lebesgue Measure
6 Generating Measures by Extension
6.1 Recap of Lebesgue and Borel Constructions
6.2 Extension Theorems
6.3 Summary - Construction of Measure Spaces
6.4 Approaches to Countable Additivity
6.5 Completion of a Measure Space
7 Finite Products of Measure Spaces
7.1 Product Space Semi-Algebras
7.2 Properties of the Semi-Algebra
7.3 Measure on the Algebra A
7.4 Extension to a Measure on the Product Space
7.5 Well-Definedness of _-Finite Product Measure Spaces
7.6 Products of Lebesgue and Borel Measure Spaces
8 Borel Measures on Rn
8.1 Rectangle Collections that Generate B(Rn)
8.2 Borel Measures and Induced Functions
8.3 Properties of General Borel Measures on Rn
9 Infinite Products of Probability Spaces
9.1 A Naive Attempt at a First Step
9.2 A Semi-Algebra A0
9.3 Finite Additivity of _A on A for Probability Spaces
9.4 Free Countable Additivity on Finite Probability Spaces
9.5 Countable Additivity on A+ in Probability Spaces on R
9.6 Extension to a Probability Measure on RN
9.7 Probability of General Rectangles
References
