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

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

## Properties

The relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem.

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

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