Pettis Theorem

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

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 $(\Omega, \mathcal)$ 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 $g : B \to \mathbb,$ the function
$g \circ Z \colon \Omega \to \mathbb \quad \text \quad \omega \mapsto g(Z(\omega))$
is a $\mathbb$-valued random variable (i.e. measurable function) in the usual sense, with respect to $\Sigma$ and the usual Borel $\sigma$-algebra on $\mathbb.$

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.

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

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References

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{{Analysis in topological vector spaces

Functional analysis
Measure theory
Types of functions