Proposition 12

— For a numerical function $f$ to belong to $\mathscr{L}^p$, it is necessary and sufficient that each of the functions $f^+$ and $f^-$ belong to $\mathscr{L}^p$.

The condition is sufficient since $f = f^+ - f^-$; it is necessary, because if $f \in \mathscr{L}^p$ then $|f| \in \mathscr{L}^p$ (Prop. 11).