Corollary 1

For every integrable function $f \in \mathscr{L}^1_F$, $|f|$ is integrable and

We shall make frequent use of Prop. 1 and its Cor. 1, on replacing $\int^* f\, d|\mu|$ or $N_1(f)$ by $\int f\, d|\mu|$ when dealing with an integrable function $\geqslant 0$. For example, for two integrable functions $f, g$ to be equivalent, it is necessary and sufficient that $\int |f - g|\, d|\mu| = 0$.

We recall that, for a function $f$ to belong to $\mathscr{L}^p_F$, it is necessary and sufficient that the function $|f|^{p-1} \cdot f$ belong to $\mathscr{L}^1_F$ (§ 3, No. 8, Cor. 1 of Th. 7), that is, that it be integrable; this is the reason for the terminology 'p-th power integrable function'. Moreover: