CHAPTER 4 - PRODUCT SPACES AND INDEPENDENCE

Summary

1. SECTION 1 introduces independence as a property that justifies some sort of factorization of probabilities or expectations. A key factorization Theorem is stated, with proof deferred to the next Section, as motivation for the measure theoretic approach. The Theorem is illustrated by a derivation of a simple form of the strong law of large numbers, under an assumption of bounded fourth moments.

2. SECTION 2 formally defines independence as a property of sigma-fields. The key Theorem from Section 1 is used as motivation for the introduction of a few standard techniques for dealing with independence. Product sigma-fields are defined.

3. SECTION 3 describes a method for constructing measures on product spaces, starting from a family of kernels.

4. SECTION 4 specializes the results from Section 3 to define product measures. The Tonelli and Fubini theorems are deduced. Several important applications are presented.

5. SECTION *5 discusses some difficulties encountered in extending the results of Sections 3 and 4 when the measures are not sigma-finite.

6. SECTION 6 introduces a blocking technique to refine the proof of the strong law of large numbers from Section 1, to get a version that requires only a second moment condition.

7. SECTION *7 introduces a truncation technique to further refine the proof of the strong law of large numbers, to get a version that requires only a first moment condition for identically distributed summands.

8. SECTION *8 discusses the construction of probability measures on products of countably many spaces.