Theorem 5

Let $(f_n)$ be an increasing sequence of functions $\geqslant 0$ in $\mathscr{L}^p$. For the upper envelope $f$ of this sequence to be p-th power integrable, it is necessary and sufficient that $\sup_n N_p(f_n) < +\infty$. The sequence $(f_n)$ is then convergent in mean of order $p$ to $f$, and

The condition being obviously necessary, we need only prove that it is sufficient. Now, if the condition is satisfied then Prop. 14 shows that the sequence $(\tilde{f}_n)$ is a Cauchy sequence in $L^p$, therefore the sequence $(f_n)$ is a Cauchy sequence in $\mathscr{L}^p$; since $f_n(x)$ tends to $f(x)$ for all $x \in X$, $f$ is p-th power integrable and is the limit of the sequence $(f_n)$ for the topology of convergence in mean of order $p$ (No. 4, Cor. 1 of Th. 3). Therefore $N_p(f_n)$ tends to $N_p(f)$ since $N_p$ is a continuous function on $\mathscr{L}^p$.