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What need (if there was any) created Riemann-Stieltjes integral? What did Riemann-Stieltjes integral want to attain?

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    $\begingroup$ I guess there was need to define integrals of the form $\int_a^b f(x)\;dg(x)$ even when $g$ is not differentiable. For example, in probability theory, to work with random variables that are not absolutely continuous. This theory was called "Stieltjes integral". Later, when the Lebesgue integral came along, that theory had two branches: Riemann-Stieltjes integral and Lebsgue-Stieltjes integral. $\endgroup$ Mar 11, 2023 at 16:31

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The Riemann-Stieljes was introduced by Thomas Stieltjes in his long paper (or book) on continued fractions,

Recherches sur les fractions continues, Mém. Sav. étr. 32, Nr. 2, 197 S. (1904).

The purpose was to unify the treatment of of continuous and discrete measures in his theory of moments.

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    $\begingroup$ Stietjes died in 1894. Did he really have a paper published 10 years later, in 1904? A long paper of his with the same title you have appeared in 1894. See eudml.org/doc/72663. A Stieltjes integral appears on the second page. $\endgroup$
    – KCd
    Mar 14, 2023 at 16:55
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    $\begingroup$ @KCd: You are right, my reference is not the first edition of this work. $\endgroup$ Mar 15, 2023 at 12:31
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An important use of the Stieltjes integral, after its creation by Stieltjes, was its use by Riesz in $1909$ to describe the continuous dual space of $C([0,1])$, the space of real-valued continuous functions $[0,1] \to \mathbf R$ equipped with the sup-norm. Riesz showed the elements of the dual space are Stieltjes integrals $f \mapsto \int_0^1 f(x)d\alpha(x)$ where $\alpha$ has bounded variation. This result was later absorbed into general measure theory: the dual space of the continuous real-valued functions on a compact Hausdorff space $X$ consists of integration of continuous functions on $X$ with respect to different Radon measures.

Mike Bertrand's website has an account of Riesz's argument here and a copy of Riesz's article here.

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