# Integration on Product Spaces

• Edwin Hewitt
• Karl Stromberg

## Abstract

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
$$\int\limits_{X \times Y} {fd\mu \times \nu = } \int\limits_X \int\limits_Y {fd\nu \ d\mu } = \int\limits_Y \int\limits_X {fd\mu \ d\nu },$$
(1)
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 $$f \in {\mathcal{S}}([a,b] \times, [c,d])$$.

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