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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.

knk
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    See here: https://math.stackexchange.com/q/380785/295791 – jnez71 May 14 '17 at 17:55
  • And here: https://math.stackexchange.com/a/2171193/295791 – jnez71 May 14 '17 at 17:55
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    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... – Did May 14 '17 at 20:48
  • This was presented by larry wasserman in his notes when talking about expectations - http://www.stat.cmu.edu/~larry/=stat705/Lecture1.pdf – knk May 17 '17 at 11:58
  • 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. – knk May 17 '17 at 12:06
  • @knk $P$ is the probability measure (not the pdf), according to those lecture notes. –  Dec 27 '19 at 12:46
  • It depends on how you define the integral with respect to such a function. One option is given in the second answer to the first question jnez71 linked; with this interpretation, the identity is tautologous. If it is supposed to be a Riemann-Stieltjes integral, then you still need to demonstrate that these two definitions are consistent. – Thorgott Dec 27 '19 at 13:05

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