Relationship between $\mathbb{E}(X)<\infty$ and $\mathbb{E}|X|<\infty$

Question

Is $\mathbb{E}(X)<\infty$ equivalent with $\mathbb{E}|X|<\infty$?

The definition of $\mathbb{E}(X)=\mathbb{E}(X^+)-\mathbb{E}(X^-)<\infty$. And this is equivalent with $\mathbb{E}|X| = \mathbb{E}(X^+)+\mathbb{E}(X^-)<\infty$. Right?

Answer 1

Rigorously (measure-wise), the expectation of a random variable $\mathbb E[X]$ can only be considered if the expectation of the absolute value $\mathbb E[|X|]$ is finite, otherwise the former is not defined, as in the Cauchy case. (See the answers to this earlier question: What makes the mean of some distributions undefined?.)

Answer 2

This gets definitional, but suppose $Z$ is a standard Cauchy random variable and $X= -|Z|$, so $X\leq 0$ and $$\lim_{c\to-\infty} E[X\{X> c\}]=-\infty.$$ I would typically say $E[X]<\infty$ but not $E[|X|]< \infty$. $E[X]$ is signed; the sign matters.

Along similar lines, I would say that $E[Z]<\infty$ is undefined but $E[|Z|]<\infty$ is false.