Probability with Measure

Chapter 1 Measure Spaces

1.1 What is measure theory?

Measure theory is the abstract mathematical theory that underlies all models of measurement of ‘size’ in the real world. This includes measurement of length, area and volume, weight and mass, and also of chance and probability. Measure theory is a branch of pure mathematics, in particular of analysis, but it plays key roles in both calculus and statistical modelling. This is because measure theory provides the foundation of both the modern theory of integration and of the modern theory of probability.

Suppose that we wish to measure the lengths of several line segments. We represent these as closed intervals of the real number line $R$ so a typical line segment is $[a, b]$ where $b > a$ . We all agree that its length is $b - a$ . We write this as

$m ([a, b]) = b - a$

and interpret this as telling us that the measure $m$ of length of the line segment $[a, b]$ is the number $b - a$ . We might also agree that if $[a_{1}, b_{1}]$ and $[a_{2}, b_{2}]$ are two non-overlapping line segments and we want to measure their combined length then we want to apply $m$ to the set-theoretic union $[a_{1}, b_{1}] \cup [a_{2}, b_{2}]$ and

$\begin{array}{rcl} m ([a_{1}, b_{1}] \cup [a_{2}, b_{2}]) & = (b_{2} - a_{2}) + (b_{1} - a_{1}) = m ([a_{1}, b_{1}]) + m ([a_{2}, b_{2}]) . \\ (1.1) \end{array}$

An isolated point $c$ has zero length and so

$m ({c}) = 0.$

If we consider the whole real line in its entirety then it has infinite length, i.e.

$m (R) = \infty .$

The key point here is that, if we try to abstract the notion of a ‘measure of length, then we should regard it as a mapping $m$ defined on subsets of the real line, that takes values in the extended non-negative real numbers $[0, \infty]$ .

We might wonder why there is any mathematical difficulty involved here, since it appears that we can easily agree on how how long a line is. The problem is that subsets of $R$ may arise naturally and still be rather complicated.

Example 1.1.1 (The Cantor Set) Let $C_{0} = [0, 1]$ . Given $C_{n}$ , define $C_{n + 1}$ by taking each sub-interval of $C_{n}$ , cutting this sub-interval into three parts of equal length and removing the open interval corresponding to the middle third. So, for each $n$ , $C_{n}$ is a set of $2^{n}$ closed intervals each of length $(\frac{1}{3})^{n}$ .

Let $C = ⋂_{n = 0}^{\infty} C_{n}$ . Clearly $C_{n + 1} \subseteq C_{n}$ , so this is a decreasing sequence of sets, and $C$ is precisely the points that ‘never end up in the middle thirds’. For example, $0 \in C_{n}$ and $\frac{1}{3} \in C_{n}$ .

The total length of the intervals in $C_{n}$ is $2^{n} (\frac{1}{3})^{n} = (\frac{2}{3})^{n}$ , which tends to zero as $n \to \infty$ . This suggests $C$ should have ‘length’ zero, but how can we make this intuition into rigorous mathematics?

The Cantor set is a ‘fractal’, which is a general term for any shape with very detailed structure. It is somewhat contrived – in fact, it was first introduced precisely as a contrived example of an odd looking shape that appeared to exist within the real line, but with no obvious purpose. Today, we know that fractal-like objects appear frequently within nature, which means that we also need to deal with them within our theory of measure.