Submitted by bovineSurveyor on

## Category:

## Description:

This course is an introduction to measure theory, especially as it applies to probability. It assumes you have previously taken a second-year undergraduate course in real analysis.

The topics covered are:

1. Definition of sigma-algebras, measurable spaces, and measures

2. Uniqueness of extension from pi-systems, CarathÃ©odory's extension theorem, Lebesgue measure on the real line

3. Independence of events and of sigma-algebras, Borel-Cantelli lemmas, Kolmogorov's zero-one law

4. Measurable functions and random variables, independence of random variables, construction of Lebesgue integral and expectation

5. Convergence in measure and convergence almost everywhere

6. Fatou's lemma, monotone and dominated convergence, differentiation under the integral

7. Product measure and Fubini's theorem

8. L^p norm, Chebyshev's, Jensen's, HÃ¶lder's and Minkowski's inequalities, uniform integrability

9. Strong law of large numbers, measure preserving maps, ergodic theorems

10. Fourier transforms, characteristic functions, weak convergence, proof of central limit theorem

These cards contain definitions, statements and proofs of theorems from the course.

I wrote these only to aid with remembering facts for the exam - I've provided them here in the hope that they'll be helpful to someone taking an equivalent course in the same position.

Since I've not included any of the explanation or motivation that accompanied the theorems, you'll be better off learning the material from a textbook if you haven't attended a similar course. I found that "Probability with Martingales" by David Williams covered a lot of the material in the course very well, and has a very good set of exercises in the back.

This may go without saying, but a fellow student has had a little trouble along these lines before: if you are taking a course in this subject, these notes are not a substitute for attending lectures.

A note on the cards: I found the LaTeX renderer used by Mnemosyne to be ugly and difficult to read, so the cards are in fact PNG images rendered from LaTeX. This does mean they are a pain to edit, and may not fit nicely on mobile screens. If anyone would like the source files to render themselves, email [email protected] and I will provide them.

## Source:

Based on my lecture notes from the Cambridge University part II course "Probability and Measure", lectured in Michaelmas term 2016 by Dr. Jason Miller

The course's official webpage is here: http://statslab.cam.ac.uk/~jpm205/teaching/mich2016/index.html

There are printed notes available on there which the lectures followed.

## Mnemosyne Compatibility:

- 2.1+