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In trying to understand some statistics material, I came across this $$\int f(x)\ dF(x) = \int f(x)\ dP(x).$$ I am not sure what this means, but with a little measure theory I have come across this looks like integration with respect to a measure.

In this case, they are saying that integrating with respect to the CDF or PDF yields the same result $E[f(x)]$.

How can the two measures be the same, are they saying the function $f(x)$ behaves the same way for increments in both the CDF and PDF? I am sorry if my math is more intuitive than algebraic.

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    $\begingroup$ See here: math.stackexchange.com/q/380785/295791 $\endgroup$ – jnez71 May 14 '17 at 17:55
  • $\begingroup$ And here: math.stackexchange.com/a/2171193/295791 $\endgroup$ – jnez71 May 14 '17 at 17:55
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    $\begingroup$ Sorry but what is your source for this? And what is $P$ supposed to mean? "In this case they are saying that integrating with respect to the CDF or PDF yields the same result " Not in this way, in any case... $\endgroup$ – Did May 14 '17 at 20:48
  • $\begingroup$ This was presented by larry wasserman in his notes when talking about expectations - stat.cmu.edu/~larry/=stat705/Lecture1.pdf $\endgroup$ – knk May 17 '17 at 11:58
  • $\begingroup$ I am guessing this might be notational abuse but it wouldn't make any sense for the two measures cdf and pdf to be same - thank you Did for clarifying that. $\endgroup$ – knk May 17 '17 at 12:06

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