I am having a bit of trouble grasping the way of calculating a Stieltjes integral where the differential term might not be at least once differentiable with respect to every variable. I have a good idea how it might work in the differentiable case, because we then have a density with respect to the differential term:
$$\frac{\partial^n F(x_1,\ldots,x_n)}{\partial x_1 \ldots \partial x_n} = f(x_1,\ldots,x_n) \implies \int_{I \subseteq \mathbb{R}^n} g(x_1,\ldots,x_n) \, dF(x_1,\ldots, x_n) = \int_{I \subseteq \mathbb{R}^n} g(x_1,\ldots,x_n) f(x_1,\ldots,x_n) \, d(x_1,\ldots, x_n) $$
and subsequent transformations can be done by equating the last expression with the equivalent iterated integral. But now I have encountered the following two statements:
$$\iint_{[0,1]^2} g(x,y) \, d^2(\min(x,y)) = \int_{0}^{1} g(x,x) \, dx \, \text{("Comonotonicity")}$$
$$\iint_{[0,1]^2} g(x,y) \, d^2(\max(0,x+y-1)) = \int_{0}^{1} g(x,1-x) \, dx \, \text{("Antimonotonicity")}$$
These are both clearly not differentiable with respect to $x$ or $y$ and trying out the method regardless leads to nonsense. I have tinkered a bit with rewriting the integral in certain ways and informally trying out some transformations, but I can't really see how it should work out that way or if I am going in the completely wrong direction. Any pointers to proofs or statements on how to do this are appreciated.
Good day!