What does $||XY||_1$ mean?

probability probability-theory measure-theory lebesgue-integral

Solution 1:

Under independence $||XY||_1=||X||_1||Y||_1$ holds even if $E|X|$ or $E|Y|$ is infinity (by Tonelli's Theorem). But $E(XY)=(EX)(EY)$ holds if the expectations are finite.

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