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 :) ).

When this definition came up in the (one and only) measure theory course I took as a student, the instructor (Peter Constantin) had this to say about it:

"It says that a set is measurable if you can make change with it."

This explanation has stuck in my mind for the last 15 years, but it is possible that I have remembered it in part because I was never really sure I understood what he meant. Anyway, it sounds good, and if I ever teach a measure theory course (I shudder to imagine the apocalyptic scenario that would necessitate my being called upon to do this: will there be any other mathematicians at all? what color will the sky be?) I might pass it along to my students.

There is an interesting exposition about the extension measure problem provided by Jun Tanaka and Peter F. McLoughlin in A Realization of Measurable Sets as Limit Points .
Abstract: Starting with a sigma finite measure on an algebra, we define a pseudometric and show how measurable sets from the Caratheodory Extension Theorem can be thought of as limit points of Cauchy sequences in the algebra.

The paper can be downloaded from arxiv at http://arxiv.org/abs/0712.2270