# 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

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

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

fis said to bealmost surely separably valued(oressentially separably valued) if there exists a subsetN⊆Xwithμ(N) = 0 such thatf(X\N) ⊆Bis separable.

Theorem(Pettis, 1938).A functionf:X→Bdefined on a measure space (X, Σ,μ) and taking values in a Banach spaceBis (strongly) measurable (with respect to Σ and the Borelσ-algebra onB) 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.

## 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.