Definition:Stieltjes Function

Definition

Let $f: \R \to \overline \R$ be a real function, where $\overline \R$ denotes the extended real numbers.

Then $f$ is said to be a Stieltjes function if and only if:

$(1): \quad f$ is increasing
$(2): \quad f$ is left-continuous.

Also known as

Some sources insist that the codomain of a Stieltjes function $f$ be $\R$.

That is, they exclude the possibility that $f$ assumes the values $\pm \infty$.

To express that a Stieltjes function $f$ does not assume infinite values, one may call $f$ a finite Stieltjes function.

Source of Name

This entry was named for Thomas Joannes Stieltjes.