# 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.,

- [math]\displaystyle{ f(t) = \lim_{n\rightarrow\infty}f_n(t)\text{ for almost every }t, \, }[/math]

where the functions [math]\displaystyle{ f_n }[/math] each have a countable range and for which the pre-image [math]\displaystyle{ f_n^{-1}(\{x\}) }[/math] is measurable for each element *x*. The concept is named after Salomon Bochner.

Bochner-measurable functions are sometimes called **strongly measurable**, **[math]\displaystyle{ \mu }[/math]-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

fisalmost surely separably valued(oressentially separably valued) if there exists a subsetN⊆Xwithμ(N) = 0 such thatf(X\N) ⊆Bis separable.

A function f :

X→Bdefined on a measure space (X, Σ,μ) and taking values in a Banach spaceBis (strongly) measurable (with respect to Σ and the Borel algebra onB) 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.

## See also

- Bochner integral
- Bochner space – Mathematical concept
- Measurable function – Function for which the preimage of a measurable set is measurable
- Measurable space – Basic object in measure theory; set and a sigma-algebra
- Pettis integral
- Vector measure
- Weakly measurable function

## References

- Showalter, Ralph E. (1997). "Theorem III.1.1".
*Monotone operators in Banach space and nonlinear partial differential equations*. Mathematical Surveys and Monographs 49. Providence, RI: American Mathematical Society. p. 103. ISBN 0-8218-0500-2. https://archive.org/details/monotoneoperatio00show..

Original source: https://en.wikipedia.org/wiki/Bochner measurable function.
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