Integrable Random Variable and Finite Expectation

Question

I would like to verify whether the following statements are equivalent:

Given a probability space $(\Omega, F, P)$, a random variable $X$ is integrable IFF it has a finite expectation IFF $E|X|<\infty$.

$E|X|<\infty\iff E(X^++X^-)<\infty\iff EX^++EX^-<\infty$.

This implies, $EX^+<\infty$ and $EX^-<\infty$, which is equivalent to have a finite expectation.

Is it correct to claim:

$X$ integrable $\iff$ $E|X|<\infty\iff$ $X$ has a finite expectation under $P$.

Answer 1

$X$ is $\mathbb P$ integrable $\iff$ $X$ is measurable and $\mathbb E X^\pm <\infty$ so $$\mathbb E X=\mathbb E X^+-X^-= \mathbb E X^+ -\mathbb E X^- <\infty $$ and $$ \mathbb E |X| =\mathbb E X^+ + X^- = \mathbb EX^+ + \mathbb E X^- <\infty .$$

And the other way around $$\mathbb E X\leq \mathbb E |X| < \infty$$ since $X^-$ is positive.