Weakly Measurable Function

Weakly measurable function

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Description:
In mathematics &mdash; specifically, in functional analysis &mdash; 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,&nbsp;&Sigma;) 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&nbsp;:&nbsp;X&nbsp;&rarr;&nbsp;B is said to be weakly measurable if, for every continuous linear functional g&nbsp;:&nbsp;B&nbsp;&rarr;&nbsp;K, the function

[itex]g circ f colon X to mathbf colon x mapsto g(f(x))[/itex]

is a measurable function with respect to &Sigma; and the usual Borel &sigma;-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&nbsp;&sube;&nbsp;X with &mu;(N)&nbsp;=&nbsp;0 such that f(X&nbsp;&nbsp;N)&nbsp;&sube;&nbsp;B is separable.</blockquote>

<blockquote>Theorem (Pettis). A function f&nbsp;:&nbsp;X&nbsp;&rarr;&nbsp;B defined on a measure space...

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