Let $x\in [a,b]$ be a real random variable with distribution $H$ that is not absolutely continuous (w.r.t Lebesgue measure). I saw this in a paper: $$ \int_a^b xH(dx) = b-\int_a^bH(x)dx. $$ I get it is integration by parts but I am not sure why it should work. Is there some implicit assumption on $H$ at work here or does this always hold? Edit: I think there may be an implicit assumption that $H$ is a Borel measure.
2026-05-16 02:37:46.1778899066
Integration by parts and non-absolutely continuous distributions
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