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.,

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

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

Bochner-measurable functions are sometimes called **strongly measurable**, **$\mu$-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.