Lemma 4

— Let $F$ be a metrizable uniform space, and let $B \subset A$ be a countable union of $\mu$-integrable sets. Then, for every Cauchy sequence $(f_n)$ in $S(A, \mu; F)$, there exists a subsequence $(f_{n_k})$ of $(f_n)$ such that $(f_{n_k}(x))$ is a Cauchy sequence in $F$ for almost every $x \in B$.

Suppose first that $B$ is integrable, and denote by $d$ a metric compatible with the uniform structure of $F$. We shall define recursively a double sequence $(f_{mn})$ of functions in $S(A, \mu; F)$ such that $f_{0n} = f_n$ for every $n$, $(f_{mn})_{n \geq 0}$ is a subsequence of $(f_{m-1,n})_{n \geq 0}$ for every $m > 0$ and, finally, such that for every $m > 0$ the set $M_{mn}$ of $x \in B$ for which $d(f_{mn}(x), f_{m,n+1}(x)) > 1/2^{m+n+1}$ has measure $|\mu|(M_{mn}) \leq 1/2^{m+n+1}$;

the possibility of such a definition follows from the fact that $(f_n)$ is a Cauchy sequence in $\mathscr{S}(A, \mu; F)$. Set $M_m = \bigcup_{n \geq 0} M_{mn}$; then

and, for every $x \in B - M_m$, we have $d(f_{mn}(x), f_{m,n+p}(x)) \leq 1/2^{m+n}$ for all $n \geq 0$ and all $p > 0$; the sequence $(f_{mn}(x))_{n \geq 0}$ is therefore a Cauchy sequence in $F$. Now let $N = \bigcap_{m=0}^{\infty} M_m$; $N$ is negligible. Set $g_n = f_{nn}$ for every $n \geq 0$; for every $m$, the sequence $(g_n)_{n \geq m}$ is a subsequence of the sequence $(f_{mn})_{n \geq 0}$; if $x \in B - N$, there exists an index $m$ such that $x \notin M_m$, which proves that the sequence $(g_n(x))$ is a Cauchy sequence in $F$.

If now $B$ is the union of a sequence $(B_m)$ of integrable sets, one can define recursively a double sequence $(g_{mn})$ such that $g_{0n} = f_n$, $(g_{mn})_{n \geq 0}$ is a subsequence of $(g_{m-1,n})_{n \geq 0}$ for every $m > 0$, and such that the sequence $(g_{mn}(x))_{n \geq 0}$ is a Cauchy sequence in $B_m - P_m$, where $P_m$ is negligible. Set $h_n = g_{nn}$ for every $n \geq 0$, so that, for every $m$, the sequence $(h_n)_{n \geq m}$ is a subsequence of $(g_{mn})_{n \geq 0}$; the sequence $(h_n(x))_{n \geq 0}$ is then a Cauchy sequence in $F$ for every $x \in B - P$, where $P = \bigcup_{m=0}^{\infty} P_m$ is negligible.