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.

If is a measurable space and is a Banach space over a field (which is the real numbers or complex numbers ), then is said to be **weakly measurable** if, for every continuous linear functional the function
is a measurable function with respect to and the usual Borel -algebra on

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 ).
Thus, as a special case of the above definition, if is a probability space, then a function is called a (-valued) **weak random variable** (or **weak random vector**) if, for every continuous linear functional the function
is a -valued random variable (i.e. measurable function) in the usual sense, with respect to and the usual Borel -algebra on

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

A function is said to be **almost surely separably valued** (or **essentially separably valued**) if there exists a subset with such that is separable.

**Theorem** (Pettis, 1938) — A function defined on a measure space and taking values in a Banach space 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 is separable, since any subset of a separable Banach space is itself separable, one can take above to be empty, and it follows that the notions of weak and strong measurability agree when is separable.

- Bochner measurable function
- Bochner integral – generalization of the Lebesgue integral to Banach-space valued functions
- Bochner space – Type of topological space
- Pettis integral
- Vector measure

- 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.

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