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Homework 1

Solve the following problems on signed measures and on Hahn and Jordan's decomposition theorems.

There was a typo (now corrected) in problem 5b, it should be \(\bigcap\) instead of \(\bigcup\).

Homework 2

Solve the following problems on the Lebesgue-Radon-Nikodym theorem and its applications.

Homework 3

Solve the following problems about the distribution function of a Borel measure on R.

Homework 4

Solve the following problems related to the Riesz-Markov-Kakutani representation theorem.

Homework 5

Solve the following problems on basic concepts in probabilities.

Homework 6

Solve the following problems on expectations and distributions.

Read the proof of the monotone class lemma (it is Lemma 2 in in Tao's Notes 2).

Homework 7

Solve the following problems on independence.

Homework 8

Solve the following problems on independence and the law of large numbers.

Homework 9

Solve the following problems on the central limit theorem.

Homework 10

Solve the following problems related to Poincaré's recurrence theorem.

Homework 11

Solve the following problems related to ergodicity and mixing.