Bochner measurable function

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In mathematics – specifically, in functional analysis – a Bochner-measurable function taking values in a Banach space is a function that equals a.e. the limit of a sequence of measurable countably-valued functions, i.e.,

f(t) = \lim_{n\rightarrow\infty}f_n(t)\text{ for almost every }t, \,

where the functions f_n each have a countable range and for which the pre-image f^{-1}\{x\} is measurable for each x. The concept is named after Salomon Bochner.

Bochner-measurable functions are sometimes called strongly measurable, \mu-measurable or just measurable (or uniformly measurable in case that the Banach space is the space of continuous linear operators between Banach spaces).

Properties[edit]

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

Function f is almost surely separably valued (or essentially separably valued) if there exists a subset N ⊆ X with μ(N) = 0 such that f(X \ N) ⊆ B is separable.

A function  : X → B defined on a measure space (X, Σ, μ) and taking values in a Banach space B is (strongly) measurable (with respect to Σ and the Borel σ-algebra on B) 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.

See also[edit]

References[edit]

  • 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. .