Corollary

— If $X$ is discrete and the measure $\mu$ is defined by a mass $+1$ at each point of $X$, then the relation $r < s$ implies $\mathscr{L}_F^r \subset \mathscr{L}_F^s$; moreover, the topology of convergence in mean of order $r$ is finer than the topology of convergence in mean of order $s$ (on $\mathscr{L}_F^r$).