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 ${\displaystyle \mathbb {R} }$ or complex numbers ${\displaystyle \mathbb {C} }$), 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

$${\displaystyle g\circ f\colon X\to \mathbb {K} \quad {\text{ defined by ))\quad x\mapsto g(f(x))}$$

${\displaystyle \Sigma }$

Borel ${\displaystyle \sigma }$-algebra

${\displaystyle \mathbb {K} .}$

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

$${\displaystyle g\circ Z\colon \Omega \to \mathbb {K} \quad {\text{ defined by ))\quad \omega \mapsto g(Z(\omega ))}$$

${\displaystyle \mathbb {K} }$

${\displaystyle \Sigma }$

${\displaystyle \sigma }$

${\displaystyle \mathbb {K} .}$

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.

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.