Lemma 8

Let $(g_n)$ be a bounded sequence in $\mathscr{L}^{\infty}(X, \mu)$ converging pointwise $\mu$-almost everywhere. Let $\widetilde{m}\in L^{\infty}(X, \mu)$ be the class of its limit. Then $\widetilde{m}_{g_n}$ converges to $\widetilde{m}_{\widetilde{g}}$ in the space $\mathscr{L}(L^p(X, \mu))$ endowed with the topology of simple convergence.

Let $f\in \mathscr{L}^p(X, \mu)$. The sequence $(g_nf)$ is bounded in $\mathscr{L}^p(X, \mu)$ and converges pointwise $\mu$-almost everywhere to $gf$. By Lebesgue's theorem (INT, IV, p. 137, § 3, n$^o7$, th. 6), the sequence $(g_nf)$ converges to $gf$ in $\mathscr{L}^p(X, \mu)$, and the assertion follows.

We shall now consider the case where $p= 2$.