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I have been recently reading up on some material and was exposed to the Lebesgue integral, although I was unsure as to why if $EX < \infty$, we have that $$EX = \int_{-\infty}^0 xdF(x) - \int_0^\infty xd[1 - F(x)].$$ I know that $$EX = \int_{-\infty}^\infty xdF(x),$$ although I was wondering is it the same as the first equation above because intuitively the "rate of change" of $d[1 - F(x)]$ is simply the same as $dF(x)$ but negative? My second question would then be suppose that $\int_{-\infty}^0 xdF(x) $ exists. Then is this really equal to $-\int_{-\infty}^0 F(x)dx $? If so, then why?

Thank you very much!

rianko
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  • These equations are called *integration by parts.* I posted a detailed analysis at https://stats.stackexchange.com/a/222497/919. – whuber Dec 21 '20 at 21:59
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    Ahh I see! Thank you very much! I was concerned because it was a Lebesgue integral, not a Riemann-Stieltjes integral, though it is good to know how the same properties connect! – rianko Dec 21 '20 at 23:19

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