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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?

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    The correct statement is rather that if $E(\xi)$ **exists** then $E(|\xi|)$ is finite -- and this is true by definition of the (Lebesgue) integral. – 2017-01-03
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    Thank you! In the book I use they say that "$E\xi$ exists or is defined if $min(E \ \xi^+, E \xi^- )< \infty$, is this the same as the definition of existance you have in mind? Would you mind showing how it follows that $E \ |\xi|$ is finite? I'm sure it's simple but I find this confusing... – 2017-01-04
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    "showing how it follows that" Sorry but did you read what I wrote? It does not "follow", this is **a definition**. – 2017-01-04
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    So the definitions are not the same? – 2017-01-04
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    Definitions (plural) of what? There is only one definition here... – 2017-01-04

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