# Bochner measurable function

In mathematics specifically, in functional analysis a Bochner-measurable function taking values in a Banach space is a function that equals a.e. 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 ${\displaystyle f^{-1}\{x\}}$ is measurable for each 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).

## Properties

The relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem.

Function f is almost surely separably valued (or essentially separably valued) if there exists a subset N  X with μ(N) = 0 such that f(X \ N)  B is separable.

A function f  : X  B defined on a measure space (X, Σ, μ) and taking values in a Banach space B is (strongly) measurable (with respect to Σ and the Borel algebra on B) if and only if it is both weakly measurable and almost surely separably valued.

In the case that B is separable, since any subset of a separable Banach space is itself separable, one can take N above to be empty, and it follows that the notions of weak and strong measurability agree when B is separable.