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Re: Defintion of measurable functions
Posted:
Dec 14, 2010 6:33 AM
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Masato Takahashi <genkimasa@gmail.com> wrote:
>>
>> (b) For every subset M of finite measure of X, f can be approximated
>> a.e. on M by a sequence of simple functions.
>
> Let us consider the following conditions.
>
> [1] f satisfies (b).
> [2] f is locally weak measurable and locally almost separably valued,
> [3] f is locally almost S-measurable and locally almost separably
> valued,
>
> It is easy to see that [1] --> [2].
>
> It can be shown by using Hahn-Banach theorem that if E is a normed
> vector space
> then [2] --> [3](for example Hille and Phillips:Functional Analysis
> and Semi-Groups).
>
> It can be shown that if E is metrizable then [3] --> [1].
>
> Therefore under the condition that E is a normed vector space, [1],
> [2], [3] are equivalent.
> I wonder if this is the case when E is a metrizable locally convex
> space.
Actually, [1]<=>[3] holds if E is metrizable, see e.g. Theorem 1.1
of my book "Integration theory - a second course".
[3]=>[2] is trivial. So your question comes down to whether a
locally weak measurable function must be locally almost S-measurable
(if necessary, one can assume also that the function is almost
separably valued, although I do not think that this plays a role here).
I do not have a copy of Hille/Phillips at hand in the moment, but isn't
it possible to just mimic the proof by using the separation theorem
for locally convex spaces (e.g. from Rudin's "Functional Analysis")
instead of Hahn-Banach?
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