Adapt the proof of Theorem 12.2 to show that any sequence ${\left(u_n\right)}_{n\in N}\subset M\left(A\right)$ with ${lim}_{n\to \infty }{u}_n\left(x\right)=u\left(x\right)$ and $\left|u_n\right|⩽g$ for some $g⩾0$ with $g^p\in L^1\left(\mu \right)$ satisfies

$$\underset{n\to \infty }{lim}\int {|u_n-u|}^{p}d\mu =0$$

[mimic the proof of Theorem 12.2 using ${|u_n-u|}^p⩽{(\left|u_n\right|+|u|)}^p⩽2^p{g}^p$.]

Data from theorem 12.2

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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