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:

$S\subseteq T\implies\mu^-(S)\le\mu^-(T)$

$\mu^-\left(\bigcup S_i\right)\ge\sum\mu^-(S_i)$

Introduction to measure

Although integration is often introduced in terms of Riemann sums, it is rather clear that Riemann integration only really represents a specific algorithm for approximating volumes based on some intuition that applies to sufficiently familiar-looking functions. In particular, it doesn't really help us in trying to understand what a volume is (the standard example is that it doesn't tell us what the volume of $\mathbb{Q}$ is).

Even without calculus, we can already calculate the volumes of polygonal shapes like lines and trapeziums through basic linear algebra -- and calculus is the science of being able to define volumes of more complicated shapes based on limiting rules from these building blocks. Riemann integration is an example of such a limiting process.

The general idea behind abstract measure theory is that we can define volume on an arbitrary space based on some "building block" sets. Essentially, if we have a set $\mathcal{F}_0$ (called a sigma basis) of shapes whose volumes we can know axiomatically, then we should be able to say that the volume of a countable union of such disjoint shapes is the sum of their volumes and the volume of the difference between a shape and its subshape is the difference between their volumes.

(These axioms look awfully similar to those of probability theory, which is why measure theory appears so much in probability. Countable unions in the definition of sigma algebras also appear more naturally in measure theory than in probability theory.)

This motivates the definition of sigma algebras in measure theory: for a space $X$, some set of its subsets $\mathcal{F}\subseteq 2^X$ is called a $\sigma$-algebra if:

• For sets $A_i\in\mathcal{F}$ ($i$ is countable), $\bigcup A_i\in\mathcal{F}$.
• For set $A\in\mathcal{F}$, $A^c\in\mathcal{F}$.

And we define a measure $\mu:\mathcal{F}\to[0,\infty]$ on a sigma algebra as a map satisfying:

• For disjoint $A_i$ ($i$ countable), $\mu\left(\bigcup A_i\right) = \sum\mu(A_i)$.

The tuple $(X,\mathcal{F})$ is called a measurable space and the tuple $(X,\mathcal{F},\mu)$ is called a measure space. The elements of the sigma algebra $\mathcal{F}$ are called measurable sets, and the basis of the sigma algebra is hopefully something simple and intuitive, like open intervals.