Properties of measure

In the first discussion we began to think about the concept of measure, with the goal being to construct a set function $m: P({\bf R}) \rightarrow {\bf R}^{\geq 0}$ that extends the notion of length to sets that are not intervals. Following the discussion questions the groups agreed upon the following properties that we would want a measure to satisfy:

1.

The measure of the empty set is 0: $m(\emptyset) = 0$.

2.

The measure of an interval is its length:

$$m(I) = \mbox{length}(I) = b - a$$

where I is an open, closed or half open interval with endpoints a < b. That is, I = (a,b), [a,b), (a,b], or [a,b].

3.

If $A \cap B = \emptyset$, then $m(A \cup B) = m(A) + m(B)$.

There were other properties involving set containment, intersection and difference that followed from the union property. For example, the union property implies that if $B \subset A$, then m(A) = m(A - B) + m(B). Since m is always non-negative, it follows that $B \subset A$ implies that $m(B) \leq m(A)$.

We also saw that the union property can be extended to arbitrary finite unions of mutually disjoint sets:

$$m(\bigcup_{i = 1}^{n} A_{i}) = \sum_{i = 1}^{n} m(A_{i})$$

This property is called finite additivity of a measure. Its proof required the use of mathematical induction. But, we also realized that we wanted to extend this further, to infinite disjoint unions because of our example of a countable set ${\bf Q} \cap [0,1]$ that we want to have measure zero. So we will add the following, more general version of additivity, called countable additivity, to our list of properties:

$$m(\bigcup_{i = 1}^{\infty} A_{i}) = \sum_{i = 1}^{\infty} m(A_{i})$$

if $A_{i} \cap A_{j} = \emptyset$ for all $i \neq j$.

A neglected property
There is one further property, called translation invariance, that seems self-evident, but will turn out to have rather subtle consequences. First, we need a definition: Given a subset $A \subset {\bf R}$ and $x \in {\bf R}$, we define $A + x = \{z: z = y + x \ \mbox{for some} y \ \in A\}$. Notice that A is a set and x is an element, but A + x is defined to be a set. Then, we will say that the measure is translation invariant if for any $A \subset {\bf R}$ and $x \in {\bf R}$, m(A + x) = m(A).