Exercise 20, § 5

Let $X$ be a compact space, $\mu$ a positive measure on $X$, and $(f_n)$ a sequence of $\mu$-measurable numerical functions. Show that the following properties are equivalent:

(i) there exists a subsequence $(f_{n_k})$ of $(f_n)$ tending to 0 almost everywhere in $X$; (ii) there exists a sequence $(\lambda_n)$ of finite real numbers such that

and such that the series with general term $\lambda_n f_n(x)$ converges almost everywhere in $X$;

(iii) there exists a sequence $(\lambda_n)$ of finite real numbers such that

and such that the series with general term $\lambda_n f_n(x)$ is absolutely convergent almost everywhere in $X$.

(To see that (i) implies (ii) and (iii), make use of Egoroff’s theorem. To see that (iii) implies (i), show that (iii) implies the existence of an increasing sequence $(A_k)$ of measurable subsets of $X$ and a subsequence $(f_{n_k})$ of $(f_n)$, such that $\mu(A_k)$ tends to $\mu(X)$ and $\int |f_{n_k} \varphi_{A_k}| d\mu$ tends to 0.)