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If $E\subset R^n$ be a Lebesgue measurable set and let $(f_k)$ be a sequence of non-negative Lebesgue measurable function on E such that $\lim f_k=f$ a.e. I want to prove that if $\int_E fd\lambda<\infty$,and$$\lim_k \int_E f_kd\lambda=\int_E fd\lambda,$$then$$\lim_k \int_A f_kd\lambda=\int_A fd\lambda$$ for every Lebesgue measurable set $A\subset E$.

Does it hold if $\displaystyle\int_E fd\lambda=\infty$?

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