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Topic: Defintion of measurable functions
Replies: 11   Last Post: Dec 18, 2010 3:43 PM

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 M. Vaeth Posts: 92 Registered: 12/13/04
Re: Defintion of measurable functions
Posted: Dec 13, 2010 8:32 AM

Masato Takahashi <genkimasa@gmail.com> wrote:
> Let X be a set.
> Let S be a sigma-algebra of subsets of X.
> Let m be a countably additive positive measure on S.
> Let K be the field of real numbers or the field of complex numbers.
> Let E be a (metrizable) topological vector space over K.
> Let f: X --> E be a map.
>
> I have been searching a good definiton of measurability of f.
> I come with the following definition.
>
> (1) For every open subset U of E, f^(-1)(U) is measurable with respect
> to m.
>
> (2) For every measurable subset M of finite measure of X,
> there are a m-null subset N of M and a countable subset H of E
> such that f(M - N) is included in the closure of H in E.

There are essentially two different definitions (although one
can also find several variations in literature):

(b) For every subset M of finite measure of X, f can be approximated
a.e. on M by a sequence of simple functions.

Property (b) is what is usually called "strong measurability" or
"Bochner measurability" and is known to be equivalent to your definition
(if E is metrizable) when you modify (1) to

(1') for every open subset U of E and every set M of finite measure the set
f^{-1}(U)\cap M belongs to the Lebesgue extension of the measure space.

(If the measure space is sigma-finite and complete, then (1) and (1') are
equivalent.)

Date Subject Author
12/12/10 Masato Takahashi
12/12/10 Gc
12/12/10 Alphy
12/12/10 Masato Takahashi
12/13/10 M. Vaeth
12/13/10 Masato Takahashi
12/14/10 M. Vaeth
12/14/10 Masato Takahashi
12/15/10 M. Vaeth
12/15/10 Masato Takahashi
12/18/10 Masato Takahashi
12/18/10 M. Vaeth