Let $\Psi: [0,\infty] \to [0,\infty]$ so that $\Psi$ is convex, and strictly increasing with $\Psi(0) = 0$ and $\Psi(\infty) = \infty.$ If $(X,A,\mu)$ is a measure space, then we define $L^{\Psi}(X,A,\mu)$ to be set of real-valued measurable functions $f$ on $X$ for which there exists $c$ such that $\int \Psi(|f|/c_0) \,\mathrm{d}\mu < \infty$.
Define $||f||_{\Psi} = \inf\left\{c > 0: \int \Psi(|f|/c)\,\mathrm{d}\mu \le 1\right\} = \min\left\{c > 0: \int \Psi(|f|/c)\,\mathrm{d}\mu \le 1\right\}.$
Is it always true that $\int \Psi\left(|f|/||f||_{\Psi}\right)\,\mathrm{d}\mu = 1$? It's clear that if there exists $c < ||f||_{\Psi}$ such that $\int\Psi(|f|/c)\,\mathrm{d}\mu<\infty,$ then this holds, so any counterexample will have to have the integral converging exactly for $c \ge ||f||_{\Psi}.$