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February 24th, 2014, 03:39 AM  #1 
Member Joined: Aug 2012 Posts: 32 Thanks: 0  Lebesgue integration question
Lebesgue integral question: Consider measure space with a summable function. I want to show that if then a.e. and therefore a.e. My proposed proof is as follows: Assume that for some . Since is positive measurable function it follows that it is the limit of an increasing sequence of simple functions which I will show as : Using Beppoi Levi I know that the limit of the integral is the integral of the limit which gives the following: This is only possible if for any we have . for all and it follows that either or . Since we assumed that is nonzero for some it follows that there is some where therefore there must be some such that and which gives for . It follows that a.e. and therefore a.e. Is this proof fine? Thanks for assistance. 

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