My study group and I were discussing this question today.
We can construct the Lebesgue measure using Caratheodory's extension theorem in the usual way:
- Given the function $F(x) = x$, we can construct a pre-measure $\mu_F$ associated with $F(x)$ defined on an algebra of "intervals" (Folland uses right-closed h-intervals);
- From this premeasure, we can induce an outer measure $\mu^*$ on the power-set of the reals;
- Using Caratheodory's extension theorem, the collection of $\mu^*$-measurable sets is a complete $\sigma$-algebra, and $\mu^*$ restricted to this $\sigma$-algebra is a complete measure.
This complete $\sigma$-algebra is in some sense a "big" structure; it is certainly larger than the Borel sets, and it must contain all of the Lebesgue measurable sets.
However, Folland provides a related Theorem:
Theorem 1.14 Let $\mathcal{A} \subset \mathcal{P}(X)$ be an algebra, $\mu_0$ a premeasure on $\mathcal{A}$, and $\mathcal{M}$ the $\sigma$-algebra generated by $\mathcal{A}$. There exists a measure $\mu$ on $\mathcal{M}$ whose restriction to $\mathcal{A}$ is $\mu_0$ -- namely, $\mu = \mu^*|_\mathcal{M}$, where $\mu^*$ is given by $$\mu^*(E) = \inf \left\{\sum_1^\infty \mu_0(A_j) : A_j \in \mathcal{A}, E \subset \bigcup_1^\infty A_j\right\}.$$
He then goes on to claim uniqueness. The proof of the theorem invokes Caratheodory, but there is a question that remains. I will try to make my thoughts clear:
- Caratheodory gives us a complete measure space, this structure is large.
- Theorem 1.14 as written tells us that we can use a premeasure on an algebra $\mathcal{A}$ to generate a measure on the $\sigma$-algebra generated by $\mathcal{A}$ -- this $\sigma$-algebra is not necessarily complete. In fact, this $\sigma$-algebra is just the Borel $\sigma$-algebra.
- We can complete the Borel $\sigma$-algebra to obtain the Lebesgue measurable sets.
However, is the completion of the Borel $\sigma$-algebra the same thing as we get by just applying Caratheodory's extension theorem to the Lebesgue outer measure to directly obtain a complete $\sigma$-algebra? Or is this a "bigger" structure than the completion of the Borel $\sigma$-algebra?