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Are there any extensions of Riemann-Stieltjes integration that are able to handle the following integral?

$\int_0^1 \alpha \space d\alpha$

where

$ \alpha(x) = \left\{ \begin{array}{lr} 0 & x < \frac{1}{2} \\ \frac{1}{2} & x = \frac{1}{2} \\ 1 & x > \frac{1}{2} \end{array} \right. $

Everything I've found so far have strong requirements about the funcions' continuity, and are undefined when there are shared discontinuities, even for such a simple function.

My interest: Where integration by parts hold.

What consequence/uses would there be to it?

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    $\begingroup$ Maybe something in one of these papers will be of use: paper 1 and paper 2 and paper 3 and paper 4 and paper 5. $\endgroup$ Commented Feb 23, 2019 at 19:50
  • $\begingroup$ @DaveL.Renfro Thanks. I'll check them later, as I'm interested in knowing many forms of integration. But at first glance it doesn't look like any of those live up to the task. Appreciated nonetheless. $\endgroup$ Commented Feb 23, 2019 at 20:24
  • $\begingroup$ The Riemann-Stieltjes integral can exist when integrand and integrator have shared discontinuity points. For example, see here. The sufficient condition for non-existence is that they both cannot be discontinuous from the right or from the left. $\endgroup$
    – RRL
    Commented Feb 23, 2019 at 20:25
  • $\begingroup$ Of course your example meets that sufficient condition. $\endgroup$
    – RRL
    Commented Feb 23, 2019 at 20:26
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    $\begingroup$ Regarding wanting to know "many forms of integration", see my answer to Dissertation on Integrals, especially the papers by Peter Bullen and Ralph Henstock (for reasons I give there). $\endgroup$ Commented Feb 24, 2019 at 7:09

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