The Riemann–Stieltjes integral is a generalization of the Riemann integral, and has a definition based on a sum analogous to the Riemann sum: $$ S(P,f,g) =\sum_{k=1}^{n} f(x_k)\Delta g(x_k) $$ where $f$ and $g$ are real valued functions, and $P$ is an arbitrary partition of the interval of integration $[a,b]$.

The Riemann integral is "measuring the area under the curve," to put it simplistically, and the Riemann sum is a reflection of this insofar as it is the approximation of that area using rectangles, and from this area definition measurability is developed. Similarly, the Lebesgue integral measures the same area as the Riemann (for functions that are Riemann integrable) and has a geometric interpretation that is straightforward and is visualized easily:

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Is there a similar geometric interpretation or intuition of what it is that the Riemann–Stieltjes integral is measuring?

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    $\begingroup$ Well, I would say that it is the area under the curve parametrised by $t \mapsto (g(t),f(t))$ computed by a Riemann rectangle method, where the rectangle are indexed by the subdivision on the argument (and without forgeting the orientation : when the curves goes to the right, the integral is counted negatively). Is that what you are looking for ? $\endgroup$ – Simon Henry Apr 13 '15 at 8:45
  • $\begingroup$ @SimonHenry Yes! That is exactly what I was looking for, thank you! I also found this question which seems to go into more depth. $\endgroup$ – Juan Sebastian Lozano Apr 13 '15 at 13:59