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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Definition

If ( X , 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 {\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 , P ) {\displaystyle (\Omega ,{\mathcal {P}})} is a probability space, then a function Z : 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 → K defined by ω ↦ g ( Z ( 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 {\displaystyle \Sigma } and the usual Borel 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 ( 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 , 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.

Definition

Properties

See also