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 $(X,\Sigma )$ is a measurable space and $B$ is a Banach space over a field $𝕂$ (which is the real numbers $ℝ$ or complex numbers $ℂ$), then $f:X\to B$ is said to be weakly measurable if, for every continuous linear functional $g:B\to 𝕂,$ the function $$g\circ f:X\to 𝕂\text{ defined by }x\mapsto g(f(x))$$ is a measurable function with respect to $\Sigma$ and the usual Borel $\sigma$-algebra on $𝕂.$

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 $B$). Thus, as a special case of the above definition, if $(\Omega ,𝒫)$ is a probability space, then a function $Z:\Omega \to B$ is called a ($B$-valued) weak random variable (or weak random vector) if, for every continuous linear functional $g:B\to 𝕂,$ the function $$g\circ Z:\Omega \to 𝕂\text{ defined by }\omega \mapsto g(Z(\omega ))$$ is a $𝕂$-valued random variable (i.e. measurable function) in the usual sense, 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.

A function $f$ is said to be almost surely separably valued (or essentially separably valued) if there exists a subset $N\subseteq X$ with $\mu (N)=0$ such that $f(X\setminus N)\subseteq B$ is separable.

Theorem (Pettis, 1938)—A function $f:X\to B$ defined on a measure space $(X,\Sigma ,\mu )$ and taking values in a Banach space $B$ 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 $B$ is separable, since any subset of a separable Banach space is itself separable, one can take $N$ above to be empty, and it follows that the notions of weak and strong measurability agree when $B$ is separable.

See also

• Bochner integral – Concept in mathematics
• Bochner space – Type of topological space