# Multidimensional Riemann integration and notion of volume or Lebesgue theory and notion of measure

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):

I quote two phrases of Walter Rudin, found in the books PMA and FA, respectively. I aggree wholeheartedly.

"It is now tempting to extend the definition of the integral over $\mathbb{R}^k$ to functions which are limits (in some sense) of continuous functions with compact support. We do not want to discuss the conditions under which this can be done; the proper setting for this question is the Lebesgue Integral."

"The validity of many important theorems of analysis depends on the completeness of the systems with which they deal. This accounts for the inadequacy of the rational number system and of the Riemann integral (...) and for the success encountered by their replacements, the real numbers and the Lebesgue Integral."

Just to take the point farther, I would say that the Riemann integral is worse than the rational numbers. We not only lack completeness, but the sets on which we can integrate are ugly and contrived, the way it deals with integrals over unbounded sets is inadequate, it lacks good properties with respect to convergence etc. Not only that, but, although the theory of Riemann integral in $\mathbb{R}$ is arguably "easier" than the Lebesgue's, this is not true for the Riemann Integral in $\mathbb{R}^n$. Just because you can reach the main definition of the integral faster, it doesn't mean that the theory is easier... and it isn't.

• Very nice answer, thank you for contribution. – luka5z Dec 20 '15 at 16:30

Riemann integration is easier to understand if you get introduced to integration theory in one dimension, cause it's closer to our intuition when it comes to calculating the area below a graph of a function in one variable.

Apart from that it can be a real pain to work with in a more general setting (several variables, general domains, integrability of limits of functions). For this the Lebesgue theory is usually much better suited, but it's more abstract. Several areas of modern mathematics would be only very hard to access without it (like functional analysis, modern theory of PDE).

I'm not sure what you are referring to by sub/superscript confusion.

Edit: getting to know the Riemann theory to some extent may help you to understand the need for a better theory, though...

• I edited my post. – luka5z Dec 20 '15 at 15:54