Stieltjes function

Summary

A Stieltjes function is a non-decreasing, right-continuous function from the real line to itself. Each Stieltjes function $F$ gives a Lebesgue–Stieltjes measure that assigns the mass $F(b) - F(a)$ to each interval $(a, b]$. Conversely, every Borel measure on $\R$ that is finite on bounded sets comes from a Stieltjes function, unique up to an additive constant. Cumulative distribution functions of random variables are the main examples.