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

Transcribed Image Text:

Given a measure space (X, A, μ). Let (fn)neN and f be extended real-valued A-measurable and integrabe functions on DE A. Suppose that lim fn = fa.e. on D. n→∞ (a) Show that if limn∞ Sp\fndu = Sp\ƒdu, then limn- fp fndu = Spfdu. (b) Show that the converse of (a) is false by constructing a counter example. Given a measure space (X, A, μ). Let (fn)neN and f be extended real-valued A-measurable and integrabe functions on DE A. Suppose that lim fn = fa.e. on D. n→∞ (a) Show that if limn∞ Sp\fndu = Sp\ƒdu, then limn- fp fndu = Spfdu. (b) Show that the converse of (a) is false by constructing a counter example.

Answer rating: 100% (QA)

The detailed ... View the full answer