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 ${\displaystyle (X,\Sigma )}$ is a measurable space and ${\displaystyle B}$ is a Banach space over a field ${\displaystyle \mathbb {K} }$ (which is the real numbers ${\displaystyle \mathbb {R} }$ or complex numbers ${\displaystyle \mathbb {C} }$), then ${\displaystyle f:X\to B}$ is said to be weakly measurable if, for every continuous linear functional ${\displaystyle g:B\to \mathbb {K} ,}$ the function $${\displaystyle g\circ f\colon X\to \mathbb {K} \quad {\text{ defined by }}\quad x\mapsto g(f(x))}$$ is a measurable function with respect to ${\displaystyle \Sigma }$ and the usual Borel ${\displaystyle \sigma }$-algebra on ${\displaystyle \mathbb {K} .}$

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

Properties

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

A function ${\displaystyle f}$ is said to be almost surely separably valued (or essentially 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.

Theorem (Pettis, 1938) — A function ${\displaystyle f:X\to B}$ defined on a measure space ${\displaystyle (X,\Sigma ,\mu )}$ and taking values in a Banach space ${\displaystyle B}$ 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 ${\displaystyle B}$ 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.

See also

• Bochner measurable function
• Bochner integral – generalization of the Lebesgue integral to Banach-space valued functionsPages displaying wikidata descriptions as a fallback
• Bochner space – Type of topological space
• Pettis integral
• Vector 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.