# 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 $$f \in {\mathcal{S}}([a,b] \times, [c,d])$$.

## Keywords

Measure Space Product Space Product Measure Compact Hausdorff Space Lebesgue Point
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.