Inmathematics—specifically, infunctional analysis—aweakly measurable function taking values in aBanach space is afunction whosecomposition with any element of thedual space is ameasurable function in the usual (strong) sense. Forseparable spaces, the notions of weak and strong measurability agree.

If${\displaystyle (X,\mathrm{\Sigma })}$ is ameasurable space and${\displaystyle B}$ is a Banach space over afield${\displaystyle \mathbb{K}}$ (which is thereal numbers${\displaystyle \mathbb{R}}$ orcomplex numbers${\displaystyle \mathbb{C}}$), then${\displaystyle f:X\to B}$ is said to beweakly measurable if, for everycontinuous linear functional${\displaystyle g:B\to \mathbb{K},}$ the function$${\displaystyle g\circ f:X\to \mathbb{K}\text{ defined by }x\mapsto g(f(x))}$$is a measurable function with respect to${\displaystyle \mathrm{\Sigma }}$ and the usualBorel${\displaystyle \sigma }$-algebra on${\displaystyle \mathbb{K}.}$

A measurable function on aprobability space is usually referred to as arandom variable (orrandom vector if it takes values in a vector space such as the Banach space${\displaystyle B}$).Thus, as a special case of the above definition, if${\displaystyle (\mathrm{\Omega },\mathcal{P})}$ is a probability space, then a function${\displaystyle Z:\mathrm{\Omega }\to B}$ is called a (${\displaystyle B}$-valued)weak random variable (orweak random vector) if, for every continuous linear functional${\displaystyle g:B\to \mathbb{K},}$ the function$${\displaystyle g\circ Z:\mathrm{\Omega }\to \mathbb{K}\text{ defined by }\omega \mapsto g(Z(\omega ))}$$is a${\displaystyle \mathbb{K}}$-valued random variable (i.e. measurable function) in the usual sense, with respect to${\displaystyle \mathrm{\Sigma }}$ and the usual Borel${\displaystyle \sigma }$-algebra on${\displaystyle \mathbb{K}.}$

The relationship between measurability and weak measurability is given by the following result, known asPettis' theorem orPettis measurability theorem.

A function${\displaystyle f}$ is said to bealmost surely separably valued (oressentially separably valued) if there exists a subset${\displaystyle N\subseteq X}$ with${\displaystyle \mu (N)=0}$ such that${\displaystyle f(X\setminus N)\subseteq B}$ is separable.

In the case that${\displaystyle B}$ is separable, since any subset of a separable Banach space is itself separable, one can take${\displaystyle N}$ above to be empty, and it follows that the notions of weak and strong measurability agree when${\displaystyle B}$ is separable.

• Bochner measurable function
• Bochner integral – Concept in mathematics
• Bochner space – Type of topological space
• Pettis integral
• Vector measure – Generalization of finite measure to Banach spaces

• 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.

• Functional analysis
• Measure theory
• Types of functions