Measure, Integration & Real Analysis · Problem 2E.13

Integration — Sheldon Axler

Mathematics — Measure Theory · borel-measurability-supremum

Problem

Prove or give a counterexample: If

f_t: \mathbf{R} \to \mathbf{R}

is a Borel measurable function for each

t \in \mathbf{R}

and

f: \mathbf{R} \to (-\infty, \infty]

is defined by

f(x) = \sup\{f_t(x) : t \in \mathbf{R}\},

then

f

is a Borel measurable function.