I'm reading about Jensen's inequality from a lecture note:
If $X$ is a random variable and $\psi: \mathbb{R} \rightarrow \mathbb{R}$ is convex and such that $\mathbb{E}(|\psi(X)|)<\infty$, then $$ \psi(\mathbb{E}(X)) \leq \mathbb{E}(\psi(X)) . $$ In particular, $|\mathbb{E}(X)| \leq \mathbb{E}(|X|)$.
It seems there is no guarantee that "$\psi(X)$ is integrable implies $X$ is integrable". As such, we don't know whether $\mathbb{E}(X)$ and thus $\psi(\mathbb{E}(X))$ are well-defined or not. Could you confirm if my understanding is correct?