Weakly measurable function

=See also=

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

:$g\; circ\; f\; colon\; X\; o\; mathbf\{K\}\; colon\; x\; mapsto\; g(f(x))$

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:

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 "σ"-algebra on "B") if and only if it is both weakly measurable and almost surely separably valued, i.e., there exists a subset "N" ⊆ "X" with "μ"("N") = 0 such that "f"("X" "N") ⊆ "B" is separable.

In the case that "B" is separable, one can take "N" to be the empty set, ∅. Hence, since any subset of a separable Banach space is itself separable, it follows that the notions of weak and strong measurability agree when "B" is separable.

ee also

* Vector-valued measure

References

* cite book
last = Showalter
first = Ralph E.
title = Monotone operators in Banach space and nonlinear partial differential equations
series = Mathematical Surveys and Monographs 49
publisher = American Mathematical Society
location = Providence, RI
year = 1997
pages = 103
isbn = 0-8218-0500-2 MathSciNet|id=1422252 (Theorem III.1.1)