# Equality in the integration with respect to a measure

Assume $$\mu$$ is a probability measure, and the associated distribution function with measure $$\mu$$ is $$F(x)$$ given by $$u((a,b])=F(b)-F(a)$$ Here, the integration of function $$g$$ with respect to the measure $$\mu$$ is denoted as $$\int g d\mu$$. In many references, it said that this integration has the following $$\int g d\mu = \int g dF.$$ I know that the notation $$d\mu$$ on left hand side implies the measure. However, since $$F$$ is not a measure, what is the meaning of $$dF$$? I am confused why the equality holds. I guess that the equality transfers the integration with respect to measure $$\mu$$ to Lebesgue Measure, but how can we interpret it?

• $\int g dF$ is defined as $\int d\mu$. In special cases it becomes a Riemann Steiltje's integral. Commented Jul 5, 2020 at 11:41

As a distribution function of a probability measure, $$F: \mathbb{R} \to \mathbb{R}$$ is non-decreasing and hence (to consider the question in a broader context) in particular of bounded variation. Now for reasonable functions $$g: \mathbb{R} \to \mathbb{R}$$ and $$h$$ of bounded variation, the term $$\int g dh$$ is (as said in the comment above) defined as the Lebesgue-Stieltjes integral of $$g$$ w.r.t. $$h$$. As I said, $$F$$ is, by assumption, of bounded variation. If you don't know much about L.-S.-integration, you may should read about this first.