Does finite variance imply on a finite mean?

Assume that a random variable has a finite variance. Does it mean it also has a finite mean?

My approach: I think since it has a finite variance, it means it is in $\mathcal{L}_2$ space of the probability measure $\mu$. Therefore, it should be in $\mathcal{L}_1$ space of the probability measure $\mu$ and thus the mean is finite.

IS that correct?

Solution 1:

You just restated the problem. To prove it, use Cauchy Schwarz Inequality: $(E|X|)^2\leq EX^2$ for any r.v. $X$.Thus $E|X|\leq (EX^2)^{1/2}<\infty$ hence you have finite mean of $|X|$. Therefore, $EX^{+}<\infty, EX^{-}<\infty$ so $EX=EX^{+}-EX^{-}$is finite.