Corollary 2

The Riesz space $L^p$ is fully lattice-ordered.

Every directed set $\mathrm{H}$ in $L^p$ (for the relation $\leq$), consisting of classes $\geq 0$ and bounded above in $L^p$, has a supremum: for, if $\tilde{h}$ is an upper bound for $\mathrm{H}$ in $L^p$, then $\|\tilde{f}\|_p \leq \|\tilde{h}\|_p$ for all $\tilde{f} \in \mathrm{H}$, and Prop. 14 is applicable. This proves the corollary (Ch. II, §1, No. 3, Prop. 1).

The conclusions of Prop. 14 no longer hold when they are formulated for the functions in $\mathscr{L}^p$ instead of their classes. To be precise, if $M$ is a subset of $\mathscr{L}^p$, consisting of functions $\geq 0$, directed for the relation $\leq$, and such that $\sup_{f \in M} N_p(f) < +\infty$, the class of the upper envelope $g$ of $M$ is not necessarily identical to the supremum in $L^p$ of the classes of the functions f \in M ; in particular, g is not necessarily p-th power integrable, and even if $g \in \mathscr{L}^p$, $N_p(g)$ can be distinct from $\sup_{f \in M} N_p(f)$ (cf. §1, No. 3, Remark 1 following Th. 3).

Nevertheless, we have the following theorem: