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.

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

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

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

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

• Bochner measurable function
• Bochner integral – Concept in mathematics
• 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.

• Functional analysis
• Measure theory
• Types of functions