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  1. Let $X$ be a real-valued random variable. Does $\mathrm{E} |X|$ finite imply $\mathrm{E} |X|^2$ finite?
  2. For a $L^p$ space wrt a probability measure, are all the $L^p, p \geq 1$ norms equivalent?

If 2 is true, then I can use 2 to show 1 is true. But my guess is that they are not true.

Thanks.

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2 Answers 2

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No. If $X$ is supported on positive integers and $P(X =n) =c/n^3$ for some normalizing constant $c >0$, then $E(X) < \infty$ and $E(X^2) =\infty$

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Hint
Your guess is correct, take a look at functions alike to $\frac1{\sqrt x}$ around $(0, a)$ and scale them so they form a random variable. You can use this counter-example for both claims.

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