Why is integration with respect to the probability measure equivalent to integration with respect to the CDF?

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.

• – jnez71 May 14 '17 at 17:55
• – jnez71 May 14 '17 at 17:55
• 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 - 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