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.

- $\mathrm{cl}:2^X\to 2^X$
- $\forall(S_i),\,\exists S,\,\bigcup\mathrm{cl}(S_i)=\mathrm{cl}(S)$
- $\forall T,\,\exists S,\,\mathrm{cl}(T)^C=\mathrm{cl}(S)$
- $S\subseteq\mathrm{cl}(S)$
- $S\subseteq\mathrm{cl}(T)\implies\mathrm{cl}(S)\subseteq\mathrm{cl}(T)$
- $\mathrm{cl}(\mathrm{cl}(S))=\mathrm{cl}(S)$
- $\mu:\mathrm{Im}(\mathrm{cl})\to[0,\infty]$
- $\mu(\bigcup S_i)=\sum\mu(S_i)$

**Caratheodory criterion**for the measurability of a set with respect to an outer measure.

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

## No comments:

## Post a Comment