This is a statement from Shiryaev's Probability that I wonder if somebody can explain to me. He starts with a definition:
Definition: We say that the expectation of $\xi$ is finite if $E \ {\xi^+} < \infty$ and $E \ {\xi^-} < \infty$.
($\xi^+ =max(\xi,0), \xi^-=-min(\xi,0)$).
Then comes the statement that I'm not sure about:
Since $|\xi|={\xi^+}+{\xi^-}$ the finiteness of $E\ \xi$, or $|E \ \xi |<\infty$ is equivalent to $E\ |\xi| <\infty$
So if I know that $E\ \xi < \infty $ then $E \ |\xi| < \infty$?
I find this a bit counter intuitive; I mean in general $\int f dx < \infty $ doesn't imply $\int |f|dx < \infty$. Why is it different in this case?