Measure, Integration & Real Analysis · Problem 5A.3

Product Measures — Sheldon Axler

Mathematics — Measure Theory · product-sigma-algebra-cross-sections-measurable-not-in-product

Problem

Let

\mathcal{B}

denote the

\sigma

-algebra of Borel subsets of

\mathbf{R}

. Show that there exists a set

E \subseteq \mathbf{R} \times \mathbf{R}

such that

[E]_a \in \mathcal{B}

and

[E]^a \in \mathcal{B}

for every

a \in \mathbf{R}

, but

E \notin \mathcal{B} \otimes \mathcal{B}