Intuition behind the definition of a measurable set

As mentioned by Yuval in the comments, this question has previously been discussed on MathOverflow. I have replicated the accepted answer by Mark below.

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 :) ).

Old question here, but it is often asked by students of measure theory.

In our text Real Analysis (Bruckner${}^2$*Thomson) Example 2.28 in Section 2.7 there is a motivation for this that Andy used in his classes.

You would have seen that the inner/outer measure idea was successful for Lebesgue measure on an interval. Certainly if you are hoping for additivity of the measure then that is an obvious idea. Besides, when Lebesgue used it the upper=lower idea was pretty much well-established.

The example Andy gives is of a very simple finite outer measure $\mu^*$ on $X=\{1,2,3\}$ with $\mu^*(X)=2$, $\mu^*(\emptyset)=0$ and otherwise $\mu^*(A)=1$. You can certainly define an inner measure but $\mu_*(A)=\mu^*(A)$ works for all subsets and yet that measure is not even finitely additive. Evidently to spot the sets on which the measure will be additive you need to test more than just whether $\mu^*(A)+\mu^*(X\setminus A)=\mu^*(X)$. Caratheodory came up with the idea of testing all sets not just $X$ itself (since it clearly doesn't work).