A definition of measure

The fifth question on the discussion sheet asked how far we can get using these properties. That is, for how large a collection of subsets of ${\bf R}$ can we define m based on these properties alone. For example, we can certainly obtain m for any finite union of disjoint intervals. Using countable additivity we can extend this to countable unions of disjoint intervals. As we argued in class, every open subset of ${\bf R}$ is a countable union of disjoint open intervals. Thus it is possible to compute the measure of open subset directly from a knowledge of the measure of open intervals and countable additivity of the measure. This is a large and interesting collection of subsets of ${\bf R}$, but it is not large enough.

In fact, we noted that the set $[0,1] - {\bf Q}$ falls outside of this collection (although we argued that as the complement in [0,1] of a countable set, it should have measure 1.) Since it is uncountable and it contains no intervals of positive length, there is no way to build up this set or its measure from that of intervals. This example shows that we cannot extend the notion of length to all sets simply by using the properties and that we are forced to define measure more generally.

Naively, we want to use the containment property above in combination with countable additivity to ``shrink wrap'' sets. Let us suppose A is a bounded set. Then A is contained in an interval $A \subset (a,b)$, so $m(A) \leq b - a$. At first, we may want to say that the measure of A should be the length of the smallest interval containing A, but that is inadequate since A might come in several pieces. This leads us to say that the measure of A should be the sum of the lengths of the ``smallest'' intervals that can be used to cover A. Unfortunately, smallest is not precise enough here, and, in fact, there is no way to specify a smallest collection. Rather for each such ``cover'', we want to look at the sum of the measures of the covering subsets. Then, we want to look at the greatest lower bound, or infimum of these values.

$$m(A) = inf \sum_{i = 1}^{\infty} \mbox{length}(A_{i})$$

where the inf is taken over all countable collections of open intervals A_i whose union contains A,

$$A \subset \bigcup_{i=1}^{\infty} A{_i}$$

So, we have a definition. Is it a good one?