Weakly measurable function
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 [math]\displaystyle{ (X, \Sigma) }[/math] is a measurable space and [math]\displaystyle{ B }[/math] is a Banach space over a field [math]\displaystyle{ \mathbb{K} }[/math] (which is the real numbers [math]\displaystyle{ \R }[/math] or complex numbers [math]\displaystyle{ \Complex }[/math]), then [math]\displaystyle{ f : X \to B }[/math] is said to be weakly measurable if, for every continuous linear functional [math]\displaystyle{ g : B \to \mathbb{K}, }[/math] the function [math]\displaystyle{ g \circ f \colon X \to \mathbb{K} \quad \text{ defined by } \quad x \mapsto g(f(x)) }[/math] is a measurable function with respect to [math]\displaystyle{ \Sigma }[/math] and the usual Borel [math]\displaystyle{ \sigma }[/math]-algebra on [math]\displaystyle{ \mathbb{K}. }[/math]
A measurable function on a probability space is usually referred to as a random variable (or random vector if it takes values in a vector space such as the Banach space [math]\displaystyle{ B }[/math]). Thus, as a special case of the above definition, if [math]\displaystyle{ (\Omega, \mathcal{P}) }[/math] is a probability space, then a function [math]\displaystyle{ Z : \Omega \to B }[/math] is called a ([math]\displaystyle{ B }[/math]-valued) weak random variable (or weak random vector) if, for every continuous linear functional [math]\displaystyle{ g : B \to \mathbb{K}, }[/math] the function [math]\displaystyle{ g \circ Z \colon \Omega \to \mathbb{K} \quad \text{ defined by } \quad \omega \mapsto g(Z(\omega)) }[/math] is a [math]\displaystyle{ \mathbb{K} }[/math]-valued random variable (i.e. measurable function) in the usual sense, with respect to [math]\displaystyle{ \Sigma }[/math] and the usual Borel [math]\displaystyle{ \sigma }[/math]-algebra on [math]\displaystyle{ \mathbb{K}. }[/math]
Properties
The relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem.
A function [math]\displaystyle{ f }[/math] is said to be almost surely separably valued (or essentially separably valued) if there exists a subset [math]\displaystyle{ N \subseteq X }[/math] with [math]\displaystyle{ \mu(N) = 0 }[/math] such that [math]\displaystyle{ f(X \setminus N) \subseteq B }[/math] is separable.
Theorem (Pettis, 1938) — A function [math]\displaystyle{ f : X \to B }[/math] defined on a measure space [math]\displaystyle{ (X, \Sigma, \mu) }[/math] and taking values in a Banach space [math]\displaystyle{ B }[/math] is (strongly) measurable (that equals a.e. the limit of a sequence of measurable countably-valued functions) if and only if it is both weakly measurable and almost surely separably valued.
In the case that [math]\displaystyle{ B }[/math] is separable, since any subset of a separable Banach space is itself separable, one can take [math]\displaystyle{ N }[/math] above to be empty, and it follows that the notions of weak and strong measurability agree when [math]\displaystyle{ B }[/math] is separable.
See also
- Bochner measurable function
- Bochner integral
- Bochner space – Mathematical concept
- Pettis integral
- Vector measure
References
- Pettis, B. J. (1938). "On integration in vector spaces". Trans. Amer. Math. Soc. 44 (2): 277–304. doi:10.2307/1989973. ISSN 0002-9947.
- 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.
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