Given a probability space $(\Omega, \mathscr{F}, P)$, fix measurable functions $f,g:X \to \mathbb{R}$ such that $f^2g$ is integrable.
Is also $fg$ integrable?
2026-05-03 12:16:21.1777810581
$f^2g$ integrable then $fg$ integrable
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The answer is no. Take my example in the comments: $X = [0,1]$ with Lebesgue measure, $g(x) = \frac 1 {x^{3/2}}$ and $f(x) = x^{1/2}$.