There seem to be a lot of in stochastic analysis of pathwise Lebesgue-Stieltjes integral generated by the quadratic covariation of two processes, $<M,N>$.
If all our functions are positive, the Lebesgue-Stieltjes integral behaves very well. But if the function only has bounded variation, it has a positive and a negative part, we get a signed measure, and it is not clear if this is well defined etc..
So I am wondering, do you know about any books, either in classical real analysis, or books regarding stochastic integration, where this subject is treated with care? When I am reading books I often run into problems with these type of pathwise integrals, because I can not see why everything works out fine, and everything is well defined.
Do you know about any books in either real analysis or stochastic analysis, where they take care of this?