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Re: Defintion of measurable functions
Posted:
Dec 13, 2010 8:32 AM
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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):
(a) Your property (1)
(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.)
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