Lebesgue-Stieltjes integration

In measure-theoretic analysis and related branches of mathematics, Lebesgue–Stieltjes integration generalizes Riemann–Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be associated to any function of bounded variation on the real line. The Lebesgue–Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of this kind.

Lebesgue–Stieltjes integrals, named for Henri Leon Lebesgue and Thomas Joannes Stieltjes, are also known as Lebesgue–Radon integrals or just Radon integrals, after Johann Radon, to whom much of the theory is due. They find common application in probability and , and in certain branches of analysis including potential theory.

The Lebesgue–Stieltjes integral

is defined when $f : [a, b] \to R$ is Borel-measurable and bounded and $g : [a, b] \to R$ is of bounded variation in $[a, b]$ and right-continuous, or when $f$ is non-negative and $g$ is monotone and right-continuous. To start, assume that $f$ is non-negative and $g$ is monotone non-decreasing and right-continuous. Define $w ((s, t]) = g (t) - g (s)$ and $w ({a}) = 0$ (Alternatively, the construction works for $g$ left-continuous, $w ([s, t)) = g (t) - g (s)$ and $w ({b}) = 0$ ).

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