"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.

A counterexample is the Cauchy distribution. It is symmetric around $0$ but both tails of the distribution are 'too' heavy:
$$E[X]=\int_{-\infty}^{\infty} x\cdot\frac{1}{\pi(1+x^2)}\,\mathrm{d}x = \int_{-\infty}^{0} x\cdot\frac{1}{\pi(1+x^2)}\,\mathrm{d} x + \int_{0}^{\infty} x\cdot\frac{1}{\pi(1+x^2)}\,\mathrm{d} x =\infty -\infty$$
