V. Product spaces and product measures

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Find these notes at gregorkb.github.io/probtheory/

Product spaces

• Define Cartesian product of two sample spaces, measurable rectangles, product \(\sigma\)-algebra, product measure space.

• Define the cross section of a set and of a function.

• Proposition 5.1.1 of A&L. Cross sections of sets are in the expected \(\sigma\)-algebras and cross-section functions are measurable.

• Measures of cross sections are measurable functions. Product measure as it arises from integrating the measure of a cross section. Follow Theorem 5.1.2 of A&L.

Product measures and integration

• Definition of product measure and product measure space.

• Tonelli’s Theorem

• Fubini’s Theorem

• Integration by parts formula. Follow Theorem 5.2.3 of A&L.

• Add some examples to the foregoing.