Weakly Measurable Function

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In mathematics — specifically, in functional analysis — a **weakly measurable function** taking values in a Banach space is a function whose composition with any element of the dual space is a measurable function in the usual (strong) sense. For separable spaces, the notions of weak and strong measurability agree.

## Definition

If (*X*, Σ) is a measurable space and *B* is a Banach space over a field **K** (usually the real numbers **R** or complex numbers **C**), then *f* : *X* → *B* is said to be **weakly measurable** if, for every continuous linear functional *g* : *B* → **K**, the function

is a measurable function with respect to Σ and the usual Borel*σ*-algebra on **K**.

## Properties

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

<blockquote>A function*f* is said to be **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.</blockquote>

<blockquote>**Theorem** (Pettis)**.** A function *f* : *X* → *B* defined on a measure space...

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- <math>g circ f colon X to mathbf colon x mapsto g(f(x))</math>

is a measurable function with respect to Σ and the usual Borel

<blockquote>A function

<blockquote>

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