Limit of L^p norm

While proving a lemma in support of a proposition in support of a theorem in a paper I’m wrapping up, I found myself needing to show that \(\lim_{p \to \infty} ||X||_p \to ||X||_\infty\) for \(X\) some random variable. A discussion on Mathematics Stack Exchange describes a proof for the more general situation where \(f\) is a measurable function on a finite measure space and \(||f||_q < \infty\) for \(q\) sufficiently large. The conclusion in this case is that \(\lim_{p \to \infty} ||f||_p = ||f||_\infty\).

A nice thing about the random variable special case is that it allows for a slightly longer but still interesting proof (a scenic route to the result?). The sketch is first to show that \(||X||_p\) is nondecreasing in \(p\) for \(p > 0\). This can be accomplished by comparing \(||X||_p\) and \(||X||_q\) for \(q > p\) using Jensen’s inequality with the convex function \(g(x) = |x|^{q / p}\). This implies that \(||X||_p \uparrow L\) for some \(L \in \mathbb{R} \cup \{ \infty \}\). It’s easy to show that \(L \leq ||X||_\infty\). It’s only a bit less easy to show that \(L \geq ||X||_\infty\), using the standard trick of comparing \(||X||_p \geq ||\, |X| \wedge M||_p\) for \(M < ||X||_\infty\), and then carefully taking the limits in \(M\) and \(p\).