Measure, Integration & Real Analysis · Problem 5A.4

Product Measures — Sheldon Axler

Mathematics — Measure Theory · product-sigma-algebra-product-function-measurable

Problem

Suppose

(X, \mathcal{S})

and

(Y, \mathcal{T})

are measurable spaces. Prove that if

f: X \to \mathbf{R}

\mathcal{S}

-measurable and

g: Y \to \mathbf{R}

\mathcal{T}

-measurable and

h: X \times Y \to \mathbf{R}

is defined by

h(x, y) = f(x)g(y)

, then

h

(\mathcal{S} \otimes \mathcal{T})

-measurable.