# Part II - Probability and Measure

## Lectured by J. Miller, Michaelmas 2016

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# Contents

- V Full version
- 0 Introduction
- 1 Measures
- 2 Measurable functions and random variables
- 2.1 Measurable functions
- 2.2 Constructing new measures
- 2.3 Random variables
- 2.4 Convergence of measurable functions
- 2.5 Tail events
- 3 Integration
- 3.1 Definition and basic properties
- 3.2 Integrals and limits
- 3.3 New measures from old
- 3.4 Integration and differentiation
- 3.5 Product measures and Fubini's theorem
- 4 Inequalities and L
^{p}spaces - 4.1 Four inequalities
- 4.2 L
^{p}spaces - 4.3 Orthogonal projection in L
^{2} - 4.4 Convergence in L
^{1}(ℙ) and uniform integrability - 5 Fourier transform
- 5.1 The Fourier transform
- 5.2 Convolutions
- 5.3 Fourier inversion formula
- 5.4 Fourier transform in L
^{2} - 5.5 Properties of characteristic functions
- 5.6 Gaussian random variables
- 6 Ergodic theory
- 7 Big theorems