Let \((X, \mathscr{A}, \mu)\) be a finite measure space. Show that every measurable \(u \geqslant 0\) with \(\int \exp (h u(x)) \mu(d x)<\infty\) for some \(h>0\) is in \(\mathcal{L}^{p}(\mu)\) for every \(p \geqslant 1\).

[check that \(|t|^{N} / N ! \leqslant e^{|t|}\) implies \(u \in \mathcal{L}^{N}, N \in \mathbb{N}\); then use Problem 13.1.]

Data from problem 13.1

Let \((X, \mathscr{A}, \mu)\) be a finite measure space and let \(1 \leqslant q