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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.