Notes on book "Measure Theory (2nd ed.) - Cohn Donald L."

Measure Thoery: Chp 5. Product Measures

Constructions

Theorem 5.1.4. Let and be -finite measure spaces. Then there is a unique measure on the -algebra such that

holds for each in and in . Furthermore, the measure under of an arbitrary set in is given by

The measure is called the product of and .

Fubini's Theorem

Proposition 5.2.1 (Tonelli’s Theorem). Let and be -finite measure spaces, and let be -measurable. Then - the function is -measurable and the function is -measurable, and - satisfies

Theorem 5.2.2 (Fubini’s Theorem). Let and be -finite measure spaces, and let be -measurable and - integrable. Then - for -almost every in the section is -integrable and for -almost every in the section is -integrable, - the functions and defined by

and

belong to and , and

• the relation

holds.

References

[1] @book{cohn2013measure, title={Measure theory}, author={Cohn, Donald L}, year={2013}, publisher={Springer} }