Nowadays, the usual way to extend a measure on an algebra of sets to a measure on a $\sigma$-algebra, the Caratheodory approach, is by using the outer measure $m^\* $ and then taking the family of all sets $A$ satisfying $m^\* (S)=m^\* (S\cap A)+m^\* (S\cap A^c)$ for every set $S$ to be the family of measurable sets. It can then be shown that this family forms a $\sigma$-algebra and $m^\*$ restricted to this family is a complete measure. The approach is elegant, short, uses only elementary methods and is quite powerful. It is also, almost universally, seen as completely unintuitive (just google "Caratheodory unintuitve" ). 


Given that the problem of extending measures is fundamental to all of measure theory, I would like to know if anyone can provide a perspective that renders the Caratheodory approach natural and intuitive.


I'm familiar with the fact that there is a topological approach to the extension problem (see [here](http://terrytao.wordpress.com/2009/01/03/254a-notes-0a-an-alternate-approach-to-the-caratheodory-extension-theorem/) or [link text](http://arxiv.org/abs/0712.2270)) for the $\sigma$-finite case due to M.H. Stone (Maharam has actually shown how to extend it to the [general case](http://purl.pt/3098/1/)), but it doesn't give much of an insight into why the Caratheodoy approach works and that is what I`m interested in here.



Demystifying the Caratheodory Approach to Measurability
Here is an argument that may give some intuition:


Assume that $m^{\*}$ is an outer measure on $X$, and let us assume furthermore that this outer measure is finite:


$m^\* (X) < \infty$


Define an "inner measure" $m\_\*$ on $X$ by


$m\_\* (E) = m^\* (X) - m^\* (E^c) $


If $m^\*$ was, say, induced from a countably additive measure defined on some algebra of sets in $X$ (like Lebesgue measure is built using the algebra of finite disjoint unions of intervals of the form $(a,b]$), then a subset of $X$ will be measurable in the sense of Caratheodory if and only if its outer measure and inner measure agree.


From this viewpoint, the construction of the measure (as well as the $\sigma$-algebra of measurable sets) is just a generalization of the natural construction of the Riemann integral on $\mathbb{R}^n$ - you try to approximate the area of a bounded set $E$ from the outside by using finitely many rectangles, and similarly from the inside, and the set is "measurable in the sense of Riemann" (or "Jordan measurable") if the best outer approximation of its area agrees with the best inner approximation of its area.


The point here (which often isn't emphasized when Riemann integration is taught for the first time) is that the concept of "inner area" is redundant and can be defined in terms of the outer area just as I did above (you take some rectangle containing the set and consider the outer measure of the complement of the set with respect to this rectangle).


Of course, Caratheodory's construction doesn't require $m^\*$ to be finite, but I still think that this gives some decent intuition for the general case (unless you think that the construction of the Riemann integral itself is not intuitive :) ).


