Does the monotonicity of $L^p$ norms holds when the norm goes to infinity?

Question

It is fine to prove that the function $f: p\rightarrow||X||_p$ for non-negative and finite $p$ is monotone using Jensen's equality by defining a convex function $\phi(x) = |x|^{q/p}, 1<p<q<\infty:$

$||X||_p=(\mathbb{E}[|X|^p])^{1/p}=(\phi(\mathbb{E}[|X|^p]))^{1/q}$

$\leq(\mathbb{E}[\phi(|X|^p)]^{1/q}=(\mathbb{E}[|X|^q])^{1/q}=||X||_q$.

However, I am completely lost when I need to prove the monotonicity for $||X||_p = \infty$.

What should I deal with?

Answer 1

You want to prove that if $1\lt p \lt q \lt \infty$, then $\lVert X\rVert_p \leqslant \lVert X\rVert_q$. We have for a fixed $R$ thzt $$\lVert X\mathbf 1\left\{\left|X\right|\lt R\right\} \rVert_p\leqslant \lVert X\mathbf 1\left\{\left|X\right|\lt R\right\} \rVert_q.$$ Then we can conclude by the monotone convergence theorem (which can be used if the limit is finite or $+\infty$).