The Stieltjes integral of $f$ w.r.t. $g$ is $$\int_0^T f(t)\,\mathrm dg(t)=\lim_{n\to \infty }\sum_{i=0}^{n-1} f(t_i)(g(x_{i+1})-g(x_i)),$$ where $t_i\in [x_i,x_{i+1}]$ and $\{x_0,...,x_n\}$ is a partition of $[0,T]$.

What does it represent concretely ? Can it be seen as an area ? I see it often, but I don't see in what this integral is worth.

For example, what could represent $$\int_0^1 x\,\mathrm d x^2 \ \ ?$$ (despite the fact that it's equal to $\int_0^1 2x^2\,\mathrm d x$)

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    $\begingroup$ it represent a "weighted sum of quantities", something similar to the mass of an object with variable density in each of it points $\endgroup$ – Masacroso Dec 31 '18 at 11:21
  • $\begingroup$ You can also look at it as a change of variable when the function is not a diffeomorphism $\endgroup$ – lcv Dec 31 '18 at 17:24
  • $\begingroup$ Roughly speaking, it can be understood as the area under the curve $\gamma(t) = (g(t), f(t))$. Indeed, if both $f$ and $g$ are $C^1$, then the Stieltjes integral $\int_{a}^{b} f(t) \, \mathrm{d}g(t)$ is the same as the line integral $\int_{\gamma} y \, \mathrm{d}x$ where $\gamma$ is defined as above. And this interpretation can be extended to the case where $g$ is non-decreasing (such as $g(t) = \lfloor t \rfloor$). $\endgroup$ – Sangchul Lee Jan 1 at 3:29
  • $\begingroup$ On the other hand, it is good to depart from the 'area-interpretation of integral' and consider integral as simply sum of weighted quantities. This greatly helps understanding other types of integrals, such as integrals in vector calculus, contour integral in complex analysis, etc. $\endgroup$ – Sangchul Lee Jan 1 at 3:30

Here is a visualization I had previously made for myself when I was trying to get a "geometric intuition" for these integrals.

I think it's useful to consider the one of the sums in the limit, $$ \sum_{i} f(x_i) \Delta g(x_i) = \sum_{i} f(x_i) (g(x_{i+1}) - g(x_{i})) $$

The first image shows $f(x)$ on $[0,1]$, and the arrows represent the contributions $f(x_i)$ for 15 $x_i$ uniformly spaced on this interval.

The second images shows $g(x)$ on $[0,1]$, and the arrows represent the contributions $\Delta g(x_i)$. Note that when $g(x)$ is more rapidly changing, the size of $\Delta g(x_i)$ is larger. This corresponds to the weighting which @Masacroso mentions in his comment. If $g(x)$ were just $x$ then each $g(x_i)$ would be the same (like in Riemann integrals).

enter image description here

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Finally, the right side of the animation shows the partial sums of the sum above, while the left shows the individual terms of these sums.

The plot is a parametric plot, where the horizontal axis is $g(x)$ and the vertical axis is $f(x)$. In this arrangement, the contribution $f(x_i)\Delta g(x_i)$ is a rectangle on the parametric plot. Note that the orientation of the rectangles on the left determines their signs. Summing up these areas, taking into account the sign (determined by if the horizontal arrow is pointing left or right), gives the value of the sum.

enter image description here

It might also help to imagine what would happen if we replaced $g(x)$ with $x$. Then the visualization would be exactly the classic visualization of Riemann sums.

  • $\begingroup$ Very nice answer, thanks a lot for the effort :) $\endgroup$ – user623855 Jan 1 at 11:45

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