Re: Question on Egorov's theorem

From: Tony (Ttiger222_at_hotmail.com)
Date: 02/19/05 

Date: Sat, 19 Feb 2005 07:31:05 -0500

"Stephen Montgomery-Smith" <stephen@math.missouri.edu> wrote in message
news:I1BRd.9854$kS6.8692@attbi_s52...
> Tony wrote:
>> I have a simple question about Egorov's theorem :
>>
>> Egorov's theorem says "Let f_n be a sequence of measurable functions
>> tsuch
>> that f_n(x) ---> f(x) a.e. and let E be a measurable subset of X such
>> that mu(E)
>> < infinity. Then for every epsilon > 0 there exists a measurable subset
>> F
>> of E with mu(F) < epsilon such that f_n --> f uniformly on E\F."
>>
>> The proof of this theorem begins : "First note that by removing a set of
>> measure zero from X, we can assume
>> WLOG that lim f_n(x) = f(x) for each x in X."
>>
>
> In other words - let E_0 be the subset of E upon which lim_n f_n(x) =
> f(x). By assumption the measure of E_0 is the same as the measure of E.
> Now prove the theorem of E_0. Then conclude the theorem must also be true
> for E

Why if you prove the theorem for E_0 may you conclude that the theorem must
also be true for E?

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