$L^p$ spaces inclusions

Question

Let $(\mathbb{R},\mathcal{F},P)$ be a proability space, given $1\le{p}\le{q}\le{\infty}$, is there any inclusion beteween $L^p$ and $L^q$?

Answer 1

Yes, by Hölder's inequality, if $q \geq p$, $$ \int |X(x)|^p \,P(\mathrm d x) \leq \left(\int |X(x)|^q \,P(\mathrm d x)\right)^{p/q} \left(\int 1 \,P(\mathrm d x)\right)^{1-p/q} $$ that is, since $P$ is a probability measure, $\mathbb{E}(|X|^p)^{1/p} \leq \mathbb{E}(|X|^q)^{1/q}$.