Whence the Caratheodory criterion?

I've always seen it formulated like this:

Let $\mu^*$ be an outer measure of a set $X$, and let $A \subset X$ be a subset. If for every subset $E \subset X$, we have
$\mu^*(E) = \mu^*(E \cap A) + \mu^*(E \cap A^c)$,
then $A$ is a $\sigma$-algebra and the restriction of $\mu^*$ to $A$ is a measure.

Apparently, the intuition is that you have to test every subset $E \subset X$, rather than just $X$, which kinda makes sense.

1. the only difference between an outer measure and measure is the requirement for countable additivity, rather than countable subadditivity, for disjoint sets.

2. we must therefore exclude at least those sets that break countable additivity

3. the Caratheodory criterion must target those sets (or maybe conversely, it only flags up "good" for those sets which don't break countable additivity)

Maybe this is the way that the criterion was first dreamt up - I need to try and think through the details a bit more.

Continuing this line of thought: suppose we want to have $E$ measurable. Then note that, for all $A$:

$(A \cap E) \cap (A \cap E^c) = \emptyset$ i.e. those sets are disjoint
$A = (A \cap E) \cup (A \cap E^c)$ i.e. they exhaust $A$

then since we want countable additivity, we require that $E$ chops up $A$ in such a way that the following is true:

$\mu^* (A) = \mu^* ( (A \cap E) \cup (A \cap E^c) = \mu^*(A \cap E)+\mu^*(A \cap E^c)$

which is the Caratheodory criterion. Maybe that's all there is to it, apart from the formal proofs that it gives a $\sigma$-algebra, and so on?

One point though: we haven't assumed that $A$ is itself measurable; does that matter?

The trick here is that we can define it in terms of our outer measure by subtracting the outer measure of the complement of the set. For a set $E \subset X$, we can define it as:

$\mu_*(E) = \mu^*(X) - \mu^*(E^c)$

Written in a form that's perhaps more familiar:

$\mu_*(E \cap X) =\mu^*(X) - \mu^*(E^c \cap X)$

Now, we'd like condition (3): that the inner measure and the outer measure agree. That is:

$\mu^*(E \cap X) =\mu^*(X) - \mu^*(E^c \cap X) \Rightarrow \mu^*(E \cap X)+\mu^*(E^c \cap X) =\mu^*(X)$

This is very close to Caratheodory. There's some handwaving to do regarding all sets rather than just the $X$ I mentioned here

That's pretty nice, and I get the general idea - that's a fairly convincing argument for how Caratheodory (or Lebesgue or whoever) came up with the criterion. However, I'm wondering about your handwaving. You are starting from:

$E \subset X$

where there is an "obvious" concept of inner measure between $E, X$, as you point out. However, more generally, don't we want to start by considering:

$A,E \subset X$

in which case, the same criterion applied here seems to require us to consider the "inner measure" between $A,E$, which doesn't seem to be (to my mind at least) a well-defined concept. For example, if we have:

$A,E \subset X, A \cap E = \emptyset$

then applying the same approach gives us:

$\mu^*(A) = \mu^*(E \cap A)+\mu^*(E^c \cap A) = 0+\mu^*(A) = \mu^*(A)$

which is content-free.

In addition, it's still not clear to me what this is really saying when $A$ itself is not a measurable set (and the Caratheodory criterion doesn't seem to require that it be one).

But, yes, it's certainly a very interesting starting point for further consideration.

Now, we'd like condition (3): that the inner measure and the outer measure agree. That is:

The most intuitive explanation I've found of the Caratheodory criterion relates it to our usual ideas of Riemann integration.

...

Full credit should go to http://mathoverflow.net/questions/34007/demystifying-the-caratheodory-approach-to-measurability where I first came upon this explanation, and it's worth reading here if what I've said isn't clear. Hope it helps!