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.
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• See also
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Definition
If ( X , Σ Σ ) {\displaystyle (X,\Sigma )} is a measurable space and B {\displaystyle B} is a Banach space over a field K {\displaystyle \mathbb {K} } (which is the real numbers R {\displaystyle \mathbb {R} } or complex numbers C {\displaystyle \mathbb {C} } ), then f : X → → B {\displaystyle f:X\to B} is said to be weakly measurable if, for every continuous linear functional g : B → → K , {\displaystyle g:B\to \mathbb {K} ,} the function g ∘ ∘ f : : X → → K defined by x ↦ ↦ g ( f ( x ) ) {\displaystyle g\circ f\colon X\to \mathbb {K} \quad {\text{ defined by }}\quad x\mapsto g(f(x))} is a measurable function with respect to Σ Σ {\displaystyle \Sigma } and the usual Borel σ {\displaystyle \sigma } -algebra on K . {\displaystyle \mathbb {K} .}
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 B {\displaystyle B} ). Thus, as a special case of the above definition, if ( Ω Ω , P ) {\displaystyle (\Omega ,{\mathcal {P}})} is a probability space, then a function Z : Ω Ω → → B {\displaystyle Z:\Omega \to B} is called a ( B {\displaystyle B} -valued) weak random variable (or weak random vector) if, for every continuous linear functional g : B → → K , {\displaystyle g:B\to \mathbb {K} ,} the function g ∘ ∘ Z : : Ω Ω → → K defined by ω ω ↦ ↦ g ( Z ( ω ω ) ) {\displaystyle g\circ Z\colon \Omega \to \mathbb {K} \quad {\text{ defined by }}\quad \omega \mapsto g(Z(\omega ))} is a K {\displaystyle \mathbb {K} } -valued random variable (i.e. measurable function) in the usual sense, with respect to Σ Σ {\displaystyle \Sigma } and the usual Borel σ σ {\displaystyle \sigma } -algebra on K . {\displaystyle \mathbb {K} .}
Properties
The relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem.
A function f {\displaystyle f} is said to be almost surely separably valued (or essentially separably valued) if there exists a subset N ⊆ ⊆ X {\displaystyle N\subseteq X} with μ μ ( N ) = 0 {\displaystyle \mu (N)=0} such that f ( X ∖ ∖ N ) ⊆ ⊆ B {\displaystyle f(X\setminus N)\subseteq B} is separable.
Theorem (Pettis, 1938)—A function f : X → → B {\displaystyle f:X\to B} defined on a measure space ( X , Σ Σ , μ μ ) {\displaystyle (X,\Sigma ,\mu )} and taking values in a Banach space B {\displaystyle B} 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 B {\displaystyle B} is separable, since any subset of a separable Banach space is itself separable, one can take N {\displaystyle N} above to be empty, and it follows that the notions of weak and strong measurability agree when B {\displaystyle B} is separable.
See also
References
• citerefpettis1938Pettis, B. J. (1938). "On integration in vector spaces". Trans. Amer. Math. Soc. 44 (2): 277–304. doi:10.2307/1989973. ISSN 0002-9947. MR 1501970.