Say $X$ is an arbitrary random variable
I have been told that $X$ has a well-defined expectation if $\mathbb E[|X|] < \infty$. Does this mean that if $\mathbb E[X]=\infty$, then $X$ does not have a well-defined expectation? Or can there be cases where $\mathbb E[|X|] < \infty$ while $\mathbb E[X]=\infty$?
I am trying to think of examples on my own, but cannot find one.
In this sense, is the statement: The expectation $\mathbb E[X]$ exists equivalent to the statement $\mathbb E[X]$ is well-defined?
If $X$ is a random variable which is always nonnegative, then its expectation is always defined, though it may be infinite.
In general, letting $X^+=\max(X,0)$ and $X^-=\max(-X,0)$, so that $X=X^+-X^-$, then both $X^+$ and $X^-$ are always nonnegative, so their expectations are well defined. There are then four cases for $EX$: $$ \begin{array}{r|cc} &E[X^+]<\infty & E[X^+]=\infty\\ \hline E[X^-]<\infty & EX\text{ is finite} & EX=+\infty\\ E[X^-]=\infty & EX=-\infty & EX\text{ is not well defined} \end{array} $$ Only in the upper left hand corner of that table do we have $E|X|<\infty$.
Whether $EX=\infty$ implies $EX$ exists or not is a matter of convention. Some people say $\lim_{x\to0}\frac1{x^2}$ does not exist, while others say this limit exists and is equal to $+\infty$. I think most probabalists agree that $\infty$ exists.
In summary, I disagree with the sentiment that $EX$ only exists when $E|X|<\infty$. For example, the St. Petersburg random variable for which $P(X=2^i)=2^{-i}$ for $i\ge 1$ has $E|X|=\infty$, and I would say that $EX$ exists and is equal to $+\infty$.