Let (X, A, ) be a measure space and let (fn)neN, (gn)neN, and f, g be...

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Let (X, A, μ) be a measure space and let (fn)neN, (gn)neN, and f, g be extended real-valued A-measurable functions on DE A. Suppose that 1. limno fn = f and limn-Inga.e. on D. 2. (gn) and g are all integrable on D and limno fp9ndμ = SD gdµ. 3. fn9n on D for every n E N. Prove that f is integrable on D and limno fp fndu = fpfdu. Let (X, A, μ) be a measure space and let (fn)neN, (gn)neN, and f, g be extended real-valued A-measurable functions on DE A. Suppose that 1. limno fn = f and limn-Inga.e. on D. 2. (gn) and g are all integrable on D and limno fp9ndμ = SD gdµ. 3. fn9n on D for every n E N. Prove that f is integrable on D and limno fp fndu = fpfdu.

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