Let {fn} be a sequence of Lebesgue-integrable functions on [0,1] such that for any Lebesgue-measurable subset E of [0,1] the sequence ∫Efn is convergent. Assume also that \lim_n f_n\equal{}f exists almost everywhere. Prove that f is integrable and \int_E f\equal{}\lim_n \int_E f_n. Is the assertion also true if E runs only over intervals but we also assume fn≥0? What happens if [0,1] is replaced by [0,\plus{}\infty) ?
J. Szucs real analysisfunctionintegrationlimitreal analysis unsolved