Prove the following result of W.H. Young [60]; among statisticians it is also known as Pratt's lemma, see J. W. Pratt [38].

Theorem (Young; Pratt). Let ${\left(f_k\right)}_k,{\left(g_k\right)}_k$ and ${\left(G_k\right)}_k$ be sequences of integrable functions on a measure space $(X,A,\mu )$. If

(a) $f_k\left(x\right)\to f\left(x\right),g_k\left(x\right)\to g\left(x\right),G_k\left(x\right)\to G\left(x\right)$ for all $x\in X$,

(b) $g_k\left(x\right)⩽f_k\left(x\right)⩽G_k\left(x\right)$ for all $k\in N$ and all $x\in X$,

(c) $\int g_{k}d\mu \to \int gd\mu$ and $\int G_{k}d\mu \to \int Gd\mu$ with $\int gd\mu$ and $\int Gd\mu$ finite,

then ${lim}_{k\to \infty }\int f_{k}d\mu =\int fd\mu$ and $\int fd\mu$ is finite.

Explain why this generalizes Lebesgue's dominated convergence theorem, Theorem 12.2 (ii).

Data from theorem 12.2

Transcribed Image Text:

(Lebesgue; dominated convergence) Let (X, A, ) be a measure space and (un)neN CL () be a sequence of functions such that (a) un(x)