Integration on Product Spaces

Suppose that (X, ℳ, μ) and (Y, $$ \mathcal{N} $$ , ν) are two measure spaces. We wish to define a product measure space $$(X \times Y,{\mathcal{M}} \times {\mathcal{N}},\mu \times \nu),$$ where $${\mathcal{M}} \times {\mathcal{N}}$$ is an appropriate σ-algebra of subsets of X × Y and μ × ν is a measure on $${\mathcal{M}} \times {\mathcal{N}}$$ for which $$\mu \times \nu (A \times B) = \mu (A) \cdot \nu (B)$$ whenever A ∈ ℳ and $$B \times {\mathcal{N}}$$ That is, we wish to generalize the usual geometric notion of the area of a rectangle. We also wish it to be true that (1) $$\int\limits_{X \times Y} {fd\mu \times \nu = } \int\limits_X \int\limits_Y {fd\nu \ d\mu } = \int\limits_Y \int\limits_X {fd\mu \ d\nu },$$ for a reasonably large class of functions f on X × Y. Thus we want a generalization of the classical formula $$\int\limits_{[a,b] \times [c,d]} {f(x,y) \ dS} = \int\limits_a^b \int\limits_c^d {f(x,y)} dy \ dx = \int\limits_c^d \int\limits_a^b {f(x,y)} dx \ dy,$$ which, as we know from elementary analysis, is valid for all functions $$f \in {\mathcal{S}}([a,b] \times, [c,d])$$ .