Product Measure Spaces and Theorems of Fubini and Tonelli

Abstract

The product X$\times Y$ of measure spaces has as its measurable sub sets, the $\sigma $-algebra generated by the products A$\times$ B measurable sub sets of X and Y. Fubini's Theorem introduced by Guido Fubini in 1907 is a result which gives conditions under which it is possible to commute a double integral. It implies that two repeated integrals of a function of two variables are equal if the function is integrable. Tonelli's Theorem is a successor of the Fubini's Theorem. The conclusion of Tonelli's theorem is identical to that of Fubini's theorem, but the assumption that $|f|$ has a finite integral is replaced by the assumption that f is non-negative.