Why does $E|X|=\infty$ imply $EX=\infty$?

Question

In Casella and Berger Example 2.2.4(Cauchy mean), in trying to show that Cauchy random variable $X$ has no mean, the authors prove that $E|X|=\infty$. Since we want to prove that $EX=\infty$, it has to be that $E|X|=\infty$ implies $EX=\infty$. I'm wondering why $E|X|=\infty$ implies $EX=\infty$? A proof or a reference are welcome. Thanks!

Answer 1

It does not imply that. But note that in Definition 2.2.1 of the same book, the author's define $Eg(X)$ not to exist if $E|g(X)| = \infty$.