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??
Follow Math Help Forum on Facebook and Google+
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 ? Hint: .
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.
View Tag Cloud