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 $$\mathbb{K}$$ (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 $$g : B \to \mathbb{K},$$ the function $$g \circ f \colon X \to \mathbb{K} \quad \text{ defined by } \quad x \mapsto g(f(x))$$ is a measurable function with respect to $$\Sigma$$ and the usual Borel $\sigma$-algebra on $$\mathbb{K}.$$

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

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

A function $$f$$ 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.

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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.