Measure, Integration & Real Analysis · Problem 5A.1

Product Measures — Sheldon Axler

Mathematics — Measure Theory · product-sigma-algebra-rectangle-membership

Problem

Suppose

(X, \mathcal{S})

and

(Y, \mathcal{T})

are measurable spaces. Prove that if

A

is a nonempty subset of

X

and

B

is a nonempty subset of

Y

such that

A \times B \in \mathcal{S} \otimes \mathcal{T}

, then

A \in \mathcal{S}

and

B \in \mathcal{T}