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.