# 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.,

where the functions each have a countable range and for which the pre-image is measurable for each *x*. The concept is named after Salomon Bochner.

Bochner-measurable functions are sometimes called **strongly measurable**, **-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 :

X→Bdefined on a measure space (X, Σ,μ) and taking values in a Banach spaceBis (strongly) measurable (with respect to Σ and the Borelσ-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
- Pettis integral
- Bochner space
- Measurable space
- Vector-valued measure
- Measurable function

## References

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