View source for Bochner measurable function

In [[Mathematics|mathematics]] &ndash; specifically, in [[Functional analysis|functional analysis]] &ndash; a '''Bochner-measurable function''' taking values in a [[Banach space]] is a [[Function (mathematics)|function]] that equals almost everywhere the limit of a sequence of measurable countably-valued functions, i.e.,
:<math>f(t) = \lim_{n\rightarrow\infty}f_n(t)\text{ for almost every }t, \, </math>

where the functions <math>f_n</math> each have a countable range and for which the pre-image <math>f_n^{-1}(\{x\})</math> is measurable for each element&nbsp;''x''.  The concept is named after [[Biography:Salomon Bochner|Salomon Bochner]].

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

==Properties==

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

<blockquote>
Function ''f'' is '''[[Almost surely|almost surely]] separably valued''' (or '''essentially separably valued''') if there exists a subset ''N''&nbsp;&sube;&nbsp;''X'' with ''&mu;''(''N'')&nbsp;=&nbsp;0 such that ''f''(''X''&nbsp;\&nbsp;''N'')&nbsp;&sube;&nbsp;''B'' is separable.
</blockquote>

<blockquote>
A function f &nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;''B'' defined on a [[Measure space|measure space]] (''X'',&nbsp;&Sigma;,&nbsp;''&mu;'') and taking values in a Banach space ''B'' is (strongly) measurable (with respect to &Sigma; and the Borel algebra on ''B'') [[If and only if|if and only if]] it is both [[Weakly measurable function|weakly measurable]] and almost surely separably valued. 
</blockquote>

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==

* {{annotated link|Bochner integral}}
* {{annotated link|Bochner space}}
* {{annotated link|Measurable function}}
* {{annotated link|Measurable space}}
* {{annotated link|Pettis integral}}
* {{annotated link|Vector measure}}
* {{annotated link|Weakly measurable function}}

==References==

* {{cite book
| last = Showalter
| first = Ralph E.
| title = Monotone operators in Banach space and nonlinear partial differential equations
| url = https://archive.org/details/monotoneoperatio00show
| url-access = limited
| series = Mathematical Surveys and Monographs 49
| publisher = American Mathematical Society
| location = Providence, RI
| year = 1997
| page = [https://archive.org/details/monotoneoperatio00show/page/n109 103]
| isbn = 0-8218-0500-2
| mr = 1422252 
| contribution = Theorem III.1.1}}.

{{Functional analysis}}

[[Category:Functional analysis]]
[[Category:Measure theory]]
[[Category:Types of functions]]

{{Sourceattribution|Bochner measurable function}}