SCHWEITZER 1975 Problem 5

Let $\{ f_n \}$ be a sequence of Lebesgue-integrable functions on $[0,1]$ such that for any Lebesgue-measurable subset $E$ of $[0,1]$ the sequence $\int_E f_n$ is convergent. Assume also that $\lim_n f_n=f$ exists almost everywhere. Prove that $f$ is integrable and $\int_E f=\lim_n \int_E f_n$ . Is the assertion also true if $E$ runs only over intervals but we also assume $f_n \geq 0 ?$ What happens if $[0,1]$ is replaced by $[0,+\infty) ?$

J. Szucs