Strongly measurable Banach-valued function

Let $(\mathrm{\Omega },\mathcal{A},\mu )$ be a measure space and $X$ a real or complex Banach space. A function $f:\mathrm{\Omega }\to X$ is strongly measurable (or Bochner measurable) if there are measurable $X$-valued simple functions $s_n$ and a measurable null set $N$ such that

${\lim }_{n\to \mathrm{\infty }}\mathrm{\parallel }{s}_n(\omega )-f(\omega )\mathrm{\parallel }=0$

for every $\omega \in \mathrm{\Omega }\setminus N$.

This is a norm-convergence condition. It is not, by definition, merely Borel measurability of $f$ or measurability of every scalar function $x^{\ast }\circ f$. Changing $f$ on a null set preserves strong measurability: enlarge $N$ by that null set and retain the same approximants.

• The zero function is strongly measurable, witnessed by the constant zero simple sequence.
• A single measurable simple function is strongly measurable, witnessed by the constant sequence $s_n=s$ and $N=\mathrm{\varnothing }$.