Bochner measurable function

In mathematics – specifically, in functional analysis – a Bochner-measurable function taking values in a Banach space is a function that equals almost everywhere the limit of a sequence of measurable countably-valued functions, i.e.,

${\displaystyle f(t)=\lim _{n\rightarrow \infty }f_{n}(t){\text{ for almost every }}t,\,}$

where the functions ${\displaystyle f_{n}}$ each have a countable range and for which the pre-image ${\displaystyle f_{n}^{-1}(\{x\})}$ is measurable for each element x. The concept is named after Salomon Bochner.

Bochner-measurable functions are sometimes called strongly measurable, ${\displaystyle \mu }$-measurable or just measurable (or uniformly measurable in case that the Banach space is the space of continuous linear operators between Banach spaces).