## Product Measures

We continue as yesterday, considering the two -finite measure spaces and , and the product measure space .

Last time we too a measurable set and defined the functions and . We also showed that

That is, for every measurable we can define the real number

I say that this function is itself a -finite measure, and that for any measurable rectangle we have . Since measurable rectangles generate the -ring , this latter condition specifies uniquely.

To see that is a measure, we must show that it is countably additive. If is a sequence of disjoint sets then we calculate

where we have used the monotone convergence theorem to exchange the sum and the integral.

We verify the -finiteness of by covering each measurable set by countably many measurable rectangles with finite-measure sides. Since the sides’ measures are finite, the measure of the rectangle itself is the product of two finite numbers, and is thus finite.

We call the measure the “product” of the measures and , and we write . We thus have a -finite measure space that we call the “cartesian product” of the spaces and .