Stieltjes function

Remarks

Example: the floor function $F(x) = \floor{x}$ is a Stieltjes function. It is non-decreasing, and on each interval $[n, n + 1)$ with $n \in \Z$ it is constant, so it is right-continuous at every point. Its Lebesgue–Stieltjes measure puts mass $1$ at each integer.

Non-example: the ceiling function $F(x) = \ceil{x}$ is non-decreasing but is not right-continuous at the integers. At $x = 0$ one has $F(0) = 0$, while $F(y) = 1$ for all $y \in (0, 1)$, so the limit from the right is $1 \ne F(0)$.

Every non-decreasing function has one-sided limits at every point, so right-continuity only fixes the value at the countably many jumps. Two Stieltjes functions give the same measure exactly when they differ by a constant. Some authors write "distribution function" for the same notion.