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

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · graph-measurability

Problem

Suppose

(X, \mathcal{S})

is a measurable space and

f : X \to \mathbf{R}

is a function. Let

\operatorname{graph}(f) \subseteq X \times \mathbf{R}

denote the graph of

f

\operatorname{graph}(f) = \{(x, f(x)) : x \in X\}.

Let

\mathcal{B}

denote the

\sigma

-algebra of Borel subsets of

\mathbf{R}

. Prove that

\operatorname{graph}(f) \in \mathcal{S} \otimes \mathcal{B}

f

is an

\mathcal{S}

-measurable function.