Corollary 1

— If a Cauchy sequence $(f_n)$ in the space $\mathscr{L}_F^p$ is such that the sequence $(f_n(x))$ converges almost everywhere to $f(x)$*, then* $f$ is p-th power integrable and the sequence $(f_n)$ converges in mean of order $p$ to $f$.

For, there exists a subsequence $(f_{n_k})$ of $(f_n)$ such that $(f_{n_k}(x))$ converges almost everywhere to $g(x)$, where $g$ is a function in $\mathscr{L}_F^p$ such that $(f_n)$ converges in mean of order $p$ to $g$. The hypotheses therefore imply that $f(x) = g(x)$ almost everywhere, whence the corollary.