In a certain textbook, I see the Cumulative Distribution Function (CDF) of a continuous random variable X defined as $$\int_{-\infty}^{x'} dp(x)$$ where p(x) is the Probability Density Function of X.
Usually, I see the CDF defined thus: $$\int_{-\infty}^{x'} p(x)dx$$ I have only ever seen and solved integrals with respect to a variable, not with respect to a function of a variable, so I don't know if those two are equivalent. Are they? Or is the first expression wrong?
I think you meant for the first one: $$\int_{-\infty}^{x^\prime} dP(x)$$ and the integral gives the $P$-measure of the set $(-\infty,x^\prime]$. If the probability measure is absolutely continuous wrt. the Lebesgue measure (so it has a pdf) then both definitions are equivalent but the first is more general as $P$ could have a discrete part or also a singular part.