Theorem 2

— Let $A$ be a set of indices, filtered by a filter $\mathfrak{F}$ with a countable base. Let $(f_\alpha)_{\alpha \in A}$ be a family of integrable functions that, with respect to the filter $\mathfrak{F}$, converge pointwise almost everywhere to a function $f$; if there exists a numerical function $g \geqslant 0$ such that $\int^* g \, d|\mu| < +\infty$ and such that $|f_\alpha(x)| \leqslant g(x)$ almost everywhere in $X$ for each $\alpha \in A$, then the function $f$ is integrable and

The theorem follows from Lebesgue’s theorem (§ 3, No. 7, Cor. of Th. 6) since, under the conditions of the statement, $f_\alpha$ converges in mean to $f$ with respect to $\mathfrak{F}$.