Given a measure space (X, A, ). Let (fn) and f be extended real-valued A- measurable...

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Given a measure space (X, A, μ). Let (fn) and f be extended real-valued A- measurable functions on D A and assume that f is real-valued a.e. on D. Suppose there exists a sequence of positive numbers (En) such that 1. Σnew En < 0. 2. Sp fn-f|du < En for every n N for some fixed p = (0, ∞). Show that the sequence (fn) converges to f a.e. on D. (Note that no integrability of fn, f, f on D is assumed). Given a measure space (X, A, μ). Let (fn) and f be extended real-valued A- measurable functions on D A and assume that f is real-valued a.e. on D. Suppose there exists a sequence of positive numbers (En) such that 1. Σnew En < 0. 2. Sp fn-f|du < En for every n N for some fixed p = (0, ∞). Show that the sequence (fn) converges to f a.e. on D. (Note that no integrability of fn, f, f on D is assumed).

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