Is the space of bounded $\psi_\infty$ Orlicz norm random variables equal to $L^\infty$?

Question

Let $\psi_\alpha(x) = \exp(x^\alpha)-1$ for $\alpha\geq 1$. Define $$ \psi_\infty(x) = \begin{cases}\infty & x>1\\1& x = 1\\ 0 & x <1 \end{cases} $$ to be such that for any $x>0$ $\psi_\infty(x) = \lim_{\alpha\to\infty}\psi_\alpha(x)$.

Let $$\lVert X\rVert_{\psi_\alpha} = \inf\{k>0\mid \mathbb{E}[\psi_\alpha(|X|/k)] \leq 1\}$$ be the Orlicz norm associated with $\psi_\alpha$.

I am curious if the set of all random variables of finite $\lVert X\rVert_{\psi_\infty}$ norm is equal to the set of all essentially bounded random variables. More generally, I am interested if there are useful notes on Orlicz norms for functions $\psi :[0,\infty)\to[0,\infty]$ that take values in the extended real numbers. My understanding is that from the point of view of convexity it is often useful to work with such functions (they often appear when taking convex conjugates), but I often see Orlicz functions defined as convex functions

$$\psi: [0,\infty)\to[0,\infty)$$

along with some other conditions, e.g. monotonicity, and having prescribed limiting behavior around $0$ and $\infty$. I'm curious how essential omitting the potential value of $\psi(x)=\infty$ is to the entire theory.

Answer 1

For each real $k>0$, \begin{equation}E\psi_\infty(|X|/k)=\infty\,P(|X|>k)+P(|X|=k) \\ =\left\{\begin{aligned}\infty\text{ if } P(|X|>k)>0,\\ P(|X|=k)\le1\text{ if } P(|X|>k)=0. \end{aligned}\right. \end{equation} So, indeed, $\|X\|_{\psi_\infty}<\infty$ iff $X$ is essentially bounded. Moreover, $\|X\|_{\psi_\infty}=\text{ess}\,\text{sup}\,|X|$.

Generally, for any non-constant nondecreasing convex function $F\colon[0,\infty)\to[0,\infty]$ such that $F(0)\le1$, the formula \begin{equation}\|X\|_F:=\inf\{t>0\colon EF(|X|/t)\le1\} \end{equation} defines a norm on the linear space, say $L_F$, of random variables (r.v.'s) $X$ on a probability space $\mathcal P$ with $\|X\|_F<\infty$. The proof of this is the same as the one in the case when $F$ is not allowed to take the value $\infty$. (If, in addition, it is assumed that $F(0+)<1$, then all bounded r.v.'s on $\mathcal P$ will be in $L_F$.)