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Miklos Schweitzer 1975_5

Source: sequence of Lebesgue-integrable functions

December 30, 2008
real analysisfunctionintegrationlimitreal analysis unsolved

Problem Statement

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