Weakly measurable function

(Redirected from Pettis' theorem)

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

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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  

Properties

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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.

See also

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References

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  • 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.