Given $X$ be a random variable, I was confusing about the concept of $E[X]<\infty$ and $E[X]$ well-definedness. Are these two concepts means the same thing??
When we say that $E[X]$ is well-defined, we basically mean the following: Define $X^+:=\max\{X,0\}$ and $X^-:=-\min\{X,0\}$ and let $$ E[X] := E[X^+] - E[X^-] $$ where $E[X^+]<\infty$ and $E[X^-]<\infty$ or only either $E[X^+]$ or $E[X^-]$ to be infinite, i.e., we are not allow to have $\infty - \infty$.
So I could say $E[X]<\infty$ implies $E[X]$ is well-defined (since $E[X^+]$ or $E[X^-]$ are finite). But not the other way around right? Since $E[X]$ can take infinite value which fails to be $E[X]<\infty$.
This looks like a case, where it depends on which source you use.
If your definition for $E[X] = E[X^+] - E[X^-]$ to be well-defined indeed is that at most one of $E[X^+]$ and $E[X^-]$ can be infinite, then it is possible for $E[X]$ to be both well-defined and infinite (e.g. if $E[X^+] = \infty$ and $E[X^-]<\infty$, then $E[X]=\infty$). So you are right.
But there will also be textbooks that say that $E[X]$ is well-defined, only when both of $E[X^+]$ and $E[X^-]$ are finite, and in that case, we must have $E[X]<\infty$.