Let be a measurable function in with .
Show that if is Lebesgue integrable for every and that exists in ,then almost every where.
Can anyone help me with this problem??
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Here is just a general idea. Assume that on some measurable set with . Then for any once is large enough. So what will this do to ?
Just to elaborate a bit on putnam120's suggestion, for k=1,2,3,..., let . Then . If then this goes to infinity as . Also, if n is even then , so will not exist in that case.
Therefore for all k and hence . But . Therefore almost everywhere.
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