Corollary

— Let $A$ be a set of indices, filtered by a filter $\mathfrak{F}$ having a countable base. If $(f_\alpha)_{\alpha \in A}$ is a family of functions in $\mathscr{L}_F^p$ that, with respect to the filter $\mathfrak{F}$, converge pointwise almost everywhere to a function $f$, and if, moreover, there exists a numerical function $g \geq 0$ such that $\int^* g^p d|\mu| < +\infty$ and $|f_\alpha(x)| \leq g(x)$ almost everywhere in $X$ for each $\alpha \in A$, then the function $f$ is $p$-th power integrable and $f_\alpha$ tends in mean of order $p$ to $f$ with respect to the filter $\mathfrak{F}$.

For, let $(A_n)$ be a decreasing countable base of $\mathfrak{F}$, and let $\alpha_n$ be any element of $A_n$; the sequence $(f_{\alpha_n})$ converges pointwise to $f$ almost everywhere in $X$, thus Th. 6 shows that $f$ is $p$-th power integrable and that $\lim_{n \to \infty} N_p(f - f_{\alpha_n}) = 0$. Since $\mathfrak{F}$ is the intersection filter of the elementary filters associated with all such sequences $(\alpha_n)$ (GT, I, §6, No. 8, Prop. 11), $\lim_{\mathfrak{F}} N_p(f - f_\alpha)$ exists and is equal to the common limit 0 of all of the sequences $(N_p(f - f_{\alpha_n}))$.