To develope measure theory without the infinity in the codomain Since I was studying measure theory at college from Rudin's, I have been feeling awful about this right-end infinity in the codomain of the measure. Is there any way of presenting Measure Theory that circumvents this oddity?
 A: Here's one possible approach. I like it because it cleanly emphasizes the structure of measure-theoretic objects in their own right, without getting bogged down in detailed constructions. It also has the property you're looking for.
The basic approach is to describe the simplest objects (e.g. finite unions of intervals, or simple functions) as some kind of metric space, normed vector space, or normed algebra, then consider the completion. I'll give an example of how to use this approach to define the Lebesgue measure on $\mathbb{R}$.
Let $X$ be those subsets of $\mathbb{R}$ which are finite unions of intervals and points. Given two sets $A,B \in X$, their symmetric difference (the set of points on which $A$ and $B$ "disagree"), $A \triangle B$, is also a finite union of intervals and points. Define $d(A,B)$ to be the sum of the lengths of the intervals in $A \triangle B$. This distance gives a pseudometric on $X$---that is, it is a metric space but for the fact that distinct members of $X$ can still have distance 0. This pseudometric space yields a metric space $X'$ if we consider equivalence classes of $X$ under the equivalence relation "has distance 0" (which, in this case, amounts to "the symmetric difference is a finite set of points"). We can then take $X''$, the completion of $X'$. We can consider $X''$ to be the metric space of Lebesgue-measurable subsets of $\mathbb{R}$ of finite measure, up to measure 0 differences. There are similarly direct constructions for the normed vector spaces of integrable/$L^p$ functions, and for the $C^\star$ algebra of essentially bounded functions.
One weakness of this method is that, if you have an actual, honest subset of $\mathbb{R}$, it is not immediately clear what the corresponding point in $X''$ should be. That information is indirectly recoverable, and it's a matter of perspective whether you want such "set-theoretic subsets" to be your principal objects, anyway: I'm perfectly content to define "the kind of thing that has a measure" to be "a point in $X''$", rather than "a subset of $\mathbb{R}$ meeting certain niceness conditions."
