I'm studing the Riemann-Stieltjes integral [itex]\int_a^b f dg[/itex] on closed intervals [itex] [a,b] [/itex] of the real line, and the natural question came to me: How would a multiple R-S integral be defined, say, on some set [itex] D \subset \mathbb{R}^2 [/itex]? Would one use some kind of two variable integrator function [itex] g(x,y)[/itex]? Or two integrator functions of a single variable [itex]g_1(x), g_2(y)[/itex]? How about surface Stieltges integrals? Are this kind of things defined and well-studied? What I kind of see is that people quicks to "avoid" the R and R-S integrals in favour of the lebesgue integral and measure theory, and so there is not much information about R-S and it's possible generalizations. (I still hadn't see lebesgue and measure theory, so I'm trying to avoid that for the moment) Thanks.
Stieltjes integrals can be used in the context of Lebesgue integration. The differential dg is then equivalent to some measure on the domain of interest. It can be generalized to multidimensional.