Does taking expected value of two sides of inequality maintain the inequality?

Question

As some operations are not necessarily valid for maintaining an inequality (such as raising both sides of $-2 < 2$ to an even power), I wondered if taking the expected value of both sides is actually fine.

So, let's say $X$ is a random variable and $E[X^2] < \infty$ holds. Now, since $|x| < 1 + x^2$ holds for all $x \in \Bbb{R}$ (easy proof), then $E[|X|] < E[1 + X^2]$

Thanks.

Answer 1

Expectation is just an integral over the underlying probability space, and integrals respect inequalities. So yes, expectations respect inequalities.