I have finished 9 chapters of "Introduction to Analysis" by Maxwell Rosenlicht (1968). The last chapter treats about "Multiple Integrals". I find the notation a bit complicated. Also, author introduces the notion of Jordan measure and proves various properties of it. It will take a considerable amount of time to understand every proof. I wonder if it wouldn't be better to just skim through main points of that chapter and read a good book on Lebesgue theory of integration (like Wilcox's one).
What do you think? A friend of mine told me that after learning and understanding the basic theory of Riemann integration (ie. 1-dimensional case) it is better to move to Lebesgue integration (avoiding sub/superscipts confusion).
Edit (this is how the standard proof in this chapter looks like, it's conceptionally not difficult, but notation is quite complicated - or maybe I don't have enough experience):