Proposition 14

Let $H$ be a subset of the Riesz space $L^p$, consisting of classes $\geq 0$ and directed for the relation $\leq$. For $H$ to have a supremum in $L^p$, it is necessary and sufficient that

The supremum of $H$ in $L^p$ is then the limit (in the Banach space $L^p$) of the section filter of $H$.

The condition is clearly necessary, since $\tilde{f} \mapsto \| \tilde{f} \|_p$ is an increasing function on the set of elements $\geq 0$ of $L^p$. To see that it is sufficient, we first observe that it implies that the image of $H$ under the mapping $\tilde{f} \mapsto \| \tilde{f} \|_p$ has a limit in $\mathbf{R}$, by the monotone limit theorem; the image of the section filter $\mathfrak{F}$ of $H$ under this mapping is therefore a base of a Cauchy filter on $\mathbf{R}$. The proof will be complete if we show that $\mathfrak{F}$ itself is a base of a Cauchy filter on $L^p$; for, $\mathfrak{F}$ will then converge in $L^p$, since $L^p$ is complete (No. 4, Th. 2), and the proposition will follow from TVS, II, § 2, No. 7, Prop. 18.

To see that $\mathfrak{F}$ is a base of a Cauchy filter, we shall make use of the following lemma: