Proposition 1

For every positive $\mu$-integrable numerical function $f$,

For, $\int f\, d|\mu|$ and $N_1(f)$ are continuous on $\mathscr{L}^1$ and are equal for every continuous function $f \geqslant 0$ with compact support; on the other hand, every function $f \geqslant 0$ in $\mathscr{L}^1$ is the limit (in the sense of convergence in mean) of a sequence of continuous functions $\geqslant 0$ with compact support (§ 3, No. 5, Prop. 11); whence the proposition.