Outer and inner measure, complete measure

Defining measure in the sense of Borel, we are only really able to define measure on the specific sets that we can find in the sigma algebra. While this might seem intuitively acceptable (we only really have a feeling for the notion of "length" for line intervals, and not some arbitrarily bizarre sets we may be able to construct), it is easy to see that we might be able to do "better" than what the Borel measure allows.

For instance, we should be able to assign "bounds" to measures on sets. If a non-measurable set is fully contained in a set of some measure, then its measure should be less or equal to that of that set. In particular if a non-measurable set is fully contained in a set of zero measure, its measure should just be equal to zero (classic examples of this are subsets of the Cantor set). 

This is the idea behind the inner and outer measures.

Informally, the idea is that the measure of a non-measurable set is bounded below by the measures of the sets it contains and above by the measures of the sets it is contained in. More formally: the inner measure is the supremum of the measures of the measurable sets contained within and the outer measure is the infimum of the measures of the measurable sets containing it. 

I.e. given a measure space $(X, \Sigma, \mu)$ we define the inner and outer measures:

$$\mu^-(S)=\sup\{\mu(T):T\in\Sigma;\,T\subseteq S\}$$

$$\mu^+(S)=\inf\{\mu(T):T\in\Sigma;\,T\supseteq S\}$$

Our goal, as previously discussed, is to be able to measure any set that can be consistently measured. The statement we wish to make is of the form "a measure is called a complete measure if all sets whose inner and outer measures agree are measurable", i.e. $\mu$ is called a complete measure if:

$$\forall S\subseteq X, \mu^+(S)=\mu^-(S)\implies S\in\Sigma$$

It is easy to show that this is equivalent to demanding that all subsets of measure-zero sets are measurable (with measure zero), which is traditionally the more common definition of a complete measure.

More generally, we may attempt to define the notion of an outer/inner measure without reference to a measure inducing it -- and in fact we see that we may construct a measure from just an outer/inner measure alone. 

Well, the outer measure of a set is supposed to represent the measure of the smallest measurable set containing it -- informally, one may suggest that an outer measure is the composition of a measure and a "closure operator". The "axioms" defining this notion are then (based directly off the axioms for sigma algebras and measures, and the notion of an outer measure):
  • $\mathrm{cl}:2^X\to 2^X$
  • $\forall(S_i),\,\exists S,\,\bigcup\mathrm{cl}(S_i)=\mathrm{cl}(S)$
  • $\forall T,\,\exists S,\,\mathrm{cl}(T)^C=\mathrm{cl}(S)$
  • $S\subseteq\mathrm{cl}(S)$
  • $S\subseteq\mathrm{cl}(T)\implies\mathrm{cl}(S)\subseteq\mathrm{cl}(T)$
  • $\mathrm{cl}(\mathrm{cl}(S))=\mathrm{cl}(S)$
  • $\mu:\mathrm{Im}(\mathrm{cl})\to[0,\infty]$
  • $\mu(\bigcup S_i)=\sum\mu(S_i)$
What sort of $\mu^+:2^X\to[0,\infty]$ can be written as $\mu^+=\mu\circ\mathrm{cl}$ so that the above conditions are satisfied? 

To axiomatise the outer measure so, we want to first find a necessary and sufficient condition for a set to be measurable under some $\mu^+$, i.e. we wish to represent the condition $\mathrm{cl}(S)=S$ purely in terms of $\mu^+$. Well, the "idea" represented by $\mathrm{cl}(S)=S$ is that $S$ contains its "boundary", so its outer measure does not overlap with that of $S^C$ -- so $S$ and $S^C$ can be used as "building blocks" for the outer measures of other sets. I.e. for an arbitrary set $A$, 

\begin{align}\mu^+(A) &= \mu(\mathrm{cl}(A)) \\&= \mu\left(\mathrm{cl}(A)\cap S\right) + \mu\left(\mathrm{cl}(A)\cap (\mathrm{cl}(S)-S)\right) + \mu\left(\mathrm{cl}(A) \cap \mathrm{cl}(S)^C\right)\\&=\mu^+\left(A\cap S\right) + \mu^+\left(A \cap S^C\right)\end{align}
This is known as the Caratheodory criterion for the measurability of a set with respect to an outer measure.

Now we can simply think of $\mathrm{cl}$ as the operator that takes a set to the smallest Caratheodory-measurable set containing it. We then ask: what conditions must $\mu^+$ satisfy so that it is a measure on the sigma algebra of Caratheodory-measurable sets? We want to be able to show that the measure of the union of disjoint Caratheodory sets is the sum of their measures.

Well, you should be able to work out the axiomatization of the outer measure then as:
  • $S\subseteq T\implies\mu^+(S)\le\mu^+(T)$
  • $\mu^+\left(\bigcup S_i\right)\le\sum\mu^+(S_i)$
And similarly for the inner measure:

  • $S\subseteq T\implies\mu^-(S)\le\mu^-(T)$
  • $\mu^-\left(\bigcup S_i\right)\ge\sum\mu^-(S_i)$
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