Adapt the proof of Theorem 12.2 to show that any sequence ( u n ) n
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Adapt the proof of Theorem 12.2 to show that any sequence (un)n∈N⊂M(A) with limn→∞un(x)=u(x) and |un|⩽g for some g⩾0 with gp∈L1(μ) satisfies
limn→∞∫|un−u|pdμ=0
[mimic the proof of Theorem 12.2 using |un−u|p⩽(|un|+|u|)p⩽2pgp.]
Data from theorem 12.2
Transcribed Image Text:
space and (un)nEN CL () be a sequence of functions such that (a) un(x) < w(x) for all neN, xe X and some wEL' (u), (b) u(x) = limno un(x) exists in R for all x = X, then u L () and we have fun-uldu= 0; Juna , d = [ lim un du= udp. = [ud. 110 (i) lim 11 (Lebesgue; dominated convergence) Let (X, A, ) be a measure (ii) lim no
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