Theorem 6 (Lebesgue)

— Let $F$ be a Banach space, $(f_n)$ a sequence of functions in $\mathscr{L}_F^p$ such that: $1^\circ$ the sequence $(f_n(x))$ converges almost everywhere to a limit $f(x) \in F$; $2^\circ$ there exists a numerical function $g \geq 0$ such that $\int^* g^p d|\mu| < +\infty$ and $|f_n(x)| \leq g(x)$ almost everywhere in $X$, for every integer $n$. Then, the function $f$ (defined almost everywhere) is $p$-th power integrable, and the sequence $(f_n)$ converges in mean of order $p$ to $f$.

Consider the ‘double’ sequence of numerical functions $g_{mn} = |f_m - f_n|$, which belong to $\mathscr{L}^p$ (No. 5, Prop. 11); by hypothesis, $\lim_{m \to \infty, n \to \infty} g_{mn}(x) = 0$ almost everywhere, and on the other hand $|g_{mn}(x)| \leq 2g(x)$ almost everywhere; applying Cor. 3 of Th. 5 of No. 6 to this double sequence,

and since $N_p(f_m - f_n) \geq 0$ this implies $\lim_{m \to \infty, n \to \infty} N_p(f_m - f_n) = 0$; in other words, the sequence $(f_n)$ is a Cauchy sequence in $\mathscr{L}_F^p$. The theorem therefore follows from Cor. 1 of Th. 3 of No. 4.