Introducing (Borel) measures via Riesz representation theorem

Question

I think, the most standard way is to introduce measures in real analysis is to define them via the usual properties like $\sigma$-additivity, etc.

However, if the students are familiar with functional analysis at that point, one can introduce (Borel) measures via the Riesz representation theorem as the dual space of the set of continuous functions, $\mathcal{C}(\Omega)$ (i.e., the theorem becomes a definition and one has to prove the standard properties of a measure.

Questions:

1. Is there a textbook dealing measure theory like this?
2. What are - beside having Riesz representation theorem - the (dis)advantages in doing so?

Answer 1

Look for "Radon integrals".

I am not sure about any advantages, but the book by Gert Pedersen: Analysis now contains a chapter (Chapter 6) on integration theory. This is I think in the style you want:

This chapter has two functions: Throughout the book it has served as an Appendix, to which the reader was referred for definitions, arguments, and results about measures and integrals. It will now serve as a functional analyst's dream of the ideal short course in measure theory. Thus, we shall develop the theory of Radon integrals on a locally compact Hausdorff space, assuming full knowledge of topology and topological vector spaces. This theory takes as point of departure an integral (a positive linear functional) on the minimal class of topologically relevant functions on $X$, namely, the class $C_c(X)$ of continuous functions with compact supports.

The author himself is however sure that you should not start measure theory this way:

In all honesty the author will admit that the reader should have had an ordinary course (however dull) in measure and integration theory in order to appreciate fully the high-tech approach here. He should also be aware that the theory is richer than the spartan exposition might lead to believe. A study of one or more of the classical areas of application, harmonic analysis, prob­ ability, potential theory, and ergodic theory, is advisable, in order to under­ stand the significance of integration theory as a cornerstone in that dread Temple of Our Worth.

Answer 2

Students should learn many versions of the development of "integral". Simultaneously, we should ask "why bother?"

A Lebesgue "formal" theory of integration, or the possibility of such a development (even on spaces without a topology) is interesting. However, mostly we don't care about such situations; the Riesz representation theorem explains most common scenarios well enough.

We want and need a notion of integral that behaves well with respect to limit processes, such as $\lim_n \int f_n = \int \lim_n f_n$, under suitable hypotheses. This will allow us to integrate limits of functions, and so on...

(This is yet another bit of propaganda against definitions that do not admit ulterior goals.)

Answer 3

The RRT approach provides a significant advantage when you are trying to define Lebesgue measure.

The standard method is to construct Lebesgue measure using exterior measure. This is a little messy. In particular, it involves proving the Caratheodory extension theorem. You then must show that integrating a nice function against this measure gives the same result as the Riemann integral. The proof of this equivalence is messy and unintuitive.

However, suppose we already have the RRT in hand. Consider the positive function $L(f)$ on $C_c(\mathbb R^n)$ given by $f\rightarrow \int f$, where the integral is the Riemann integral. By the RRT, this gives you a measure, and it is clear by construction that integration against this measure gives the same result as the Riemann integral for all suitably nice functions. Simple!