Definition:Stieltjes Function
(Redirected from Definition:Finite Stieltjes Function)
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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.
Also see
- Results about Stieltjes functions can be found here.
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 7$: Problem $9$