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, Σ) is a measurable space and B is a Banach space over a field K (usually the real numbers R or complex numbers C), then f : X → B is said to be weakly measurable if, for every continuous linear functional g : B → K, the function

${\displaystyle g\circ f\colon X\to \mathbf {K} \colon x\mapsto g(f(x))}$

is a measurable function with respect to Σ and the usual Borel σ-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 (Ω, Σ, P) is a probability space, then a function Z: : Ω → B is called a (B-valued) weak random variable (or weak random vector) if, for every continuous linear functional g : B → K, the function

${\displaystyle g\circ Z\colon \Omega \to \mathbf {K} \colon \omega \mapsto g(Z(\omega ))}$

is a K-valued random variable (i.e. measurable function) in the usual sense, with respect to Σ and the usual Borel σ-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 ⊆ X with μ(N) = 0 such that f(X \ N) ⊆ B is separable.

Theorem (Pettis, 1938). A function f : X → B defined on a measure space (X, Σ, μ) 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 measurable function
• Bochner integral
• Pettis integral
• Vector-valued measure

References

• Pettis, B. J. (1938). "On integration in vector spaces". Trans. Amer. Math. Soc. 44 (2): 277–304. doi:10.2307/1989973. ISSN 0002-9947. MR 1501970.
• Showalter, Ralph E. (1997). "Theorem III.1.1". Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs 49. Providence, RI: American Mathematical Society. p. 103. ISBN 0-8218-0500-2. MR 1422252.