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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 $\mathbb {R}$ or complex numbers $\mathbb {C}$ ), 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.

Theorem (Pettis, 1938) — A function $f:X\to B$ defined on a measure space $(X,\Sigma ,\mu )$ 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.

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.

Analysis in topological vector spaces
Basic concepts	Abstract Wiener space Classical Wiener space Bochner space Convex series Cylinder set measure Infinite-dimensional vector function Matrix calculus Vector calculus
Derivatives	Differentiable vector–valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet derivative Total Functional derivative Gateaux derivative Directional Generalizations of the derivative Hadamard derivative Holomorphic Quasi-derivative
Measurability	Besov measure Cylinder set measure Canonical Gaussian Classical Wiener measure Measure like set functions infinite-dimensional Gaussian measure Projection-valued Vector Bochner / Weakly / Strongly measurable function Radonifying function
Integrals	Bochner Direct integral Dunford Gelfand–Pettis/Weak Regulated Paley–Wiener
Results	Cameron–Martin theorem Inverse function theorem Nash–Moser theorem Feldman–Hájek theorem No infinite-dimensional Lebesgue measure Sazonov's theorem Structure theorem for Gaussian measures
Related	Crinkled arc Covariance operator
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus
Applications	Banach manifold (bundle) Convenient vector space Choquet theory Fréchet manifold Hilbert manifold

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Definition

Properties

See also

References