Orlicz space

Remarks

Example: for $1 \le p < \infty$, $\Phi(t) = t^p$ is a Young function. The condition $\int (\abs{f}/\lambda)^p \, d\mu \le 1$ means $\lambda \ge \norm{f}_p$. So $L^\Phi = L^p$ with $\norm{f}_\Phi = \norm{f}_p$.

Non-example: $\Phi(t) = \sqrt{t}$ is not a Young function, because it is not convex. On $[0, 1]$ with Lebesgue measure the Luxemburg functional is then $\left( \int \abs{f}^{1/2} \, d\mu \right)^2$. It fails the triangle inequality: the indicator functions of $[0, \tfrac12]$ and $(\tfrac12, 1]$ each get the value $\tfrac14$, but their sum gets the value $1$.

A function $f$ lies in $L^\Phi$ exactly when $\int_X \Phi(\abs{f}/\lambda) \, d\mu < \infty$ for some $\lambda > 0$. With the Luxemburg norm, $L^\Phi(\mu)$ is a Banach space. Some authors allow Young functions with values in $[0, \infty]$; this includes $L^\infty$ as an Orlicz space.