Lebesgue–Stieltjes measure

Summary

A Lebesgue–Stieltjes measure is a measure on the real line built from a non-decreasing, right-continuous function $F$, so that the interval $(a, b]$ gets measure $F(b) - F(a)$. Every locally finite Borel measure on $\R$ arises this way. For $F(x) = x$ it is Lebesgue measure, and for the distribution function of a random variable it is the law of that variable. Integration against it is Lebesgue–Stieltjes integration, which extends the Riemann–Stieltjes integral.