"Expectation exists" vs "Expectation is finite" Are the statements


*

*"Expectation exists"

*"Expectation is finite"


equivalent? If not, could someone please provide a counterexample.
In case it's relevant, I don't know measure theory, but am confortable with probability theory & statistics at the undergraduate level.

Contra Stan Tendijck's answer, the Cauchy distribution's mean doesn't exist because $\infty-\infty$ is undefined (technically, it's an indeterminate form). However, the modulus of a Cauchy-distributed variable has a mean, but that mean is infinite. So no, a mean existing isn't equivalent to its being finite. In fact whenever $X$ has finite mean but infinite variance it's because $X^2$ has infinite mean. An example is when $X=\sqrt{|Y|}$ with a Cauchy $Y$.
