Proposition 20

Let $(f_n)_{n\in\mathbf{N}}$ be a sequence of measurable mappings of X into a metrisable space F. Suppose that the limit of $f_n(x)$ exists in the complement of a $\mu$-negligible subset of X. There exists a sequence $(C_m)_{m\in\mathbf{N}}$ of compact subsets of X whose union $\widetilde{X}$ satisfies $|\mu|(X-\widetilde{X}) = 0$, such that the functions $f_n$ are continuous on $C_m$ for all $n\in \mathbf{N}$ and all $m\in \mathbf{N}$, and the sequence $(f_n)_{n\in\mathbf{N}}$ converges uniformly to $f$ on $C_m$ for all $m\in \mathbf{N}$.

It follows from theorem 2 of INT, IV, p. 175, § 5, n$^o4$ and proposition 12, b) of INT, IV, p. 188, § 5, n$^o8$, that the set of compact subsets C of X such that the functions $f_n$ are continuous on C for all $n$, and that $(f_n)$ converges uniformly to $f$ on C, is $\mu$-dense in X (INT, IV, p. 189, § 5, n$^o8$, definition 6). The assertion then follows from the remark in INT, IV, p. 189, §5, n$^o8$.

Let N be a function belonging to $\mathscr{L}^2(X\times X)$ and let $u_N$ be the Hilbert-Schmidt mapping of $L^2(X)$ into $L^2(X)$ with kernel N (proposition 19 of IV, p. 173). Since the mapping $u_N$ is compact, there exist by proposition 9 of IV, p. 156 an element M of $\mathbf{N}$ and, denoting by I the set of integers $n\in \mathbf{N}$ such that $n\leqslant M$, orthonormal families $(f_i)_{i\in I}$ and $(g_i)_{i\in I}$ in $\mathscr{L}^2(X)$ and a family $(\alpha_i)_{i\in I}$ in $\mathbf{R}^*_+$ such that

for all $f\in L^2$(X), where the series converges in $L^2(X)$.

For each $i\in I$, let $h_i\in \mathscr{L}^2(X\times X)$ denote the function defined by $h_i(x, y) =f_i(x)g_i(y)$, so that $\widetilde{h}_i$ is the class of $f_i\otimes g_i$. By remark 9 of IV, p. 173, the series with general term $\alpha_ih_i$ converges to N in $L^2(X\times X)$.

In the remainder of this number, we suppose that N is a continuous function.