Probability with Measure

1.4 The Borel \(\sigma \)-field

In this section introduce another example of a measure space, which will represent the notion of measuring the ‘length’ of subsets of \(\R \). For an interval \([a,b]\) it is clear that the length should be \(b-a\), but as we saw in Section 1.1 for more complicated subsets of \(\R \) the situation is not so clear.

Example 1.4.1 Consider, for example, the irrational numbers \(\mathbb {I}\) and the rational numbers \(\Q \). Both \(\mathbb {I}\) and \(\Q \) are found throughout \(\R \), but they are both full of tiny holes. Can we find a meaningful way to decide what is the ‘length’ of \(\mathbb {I}\) and \(\Q \)? In fact, we will see that we can – to come in Section 1.5. But in Section 1.6 we will also show that it is possible to construct subsets of \(\R \) for which there is no meaningful idea of length.

In this section we take \(S=\R \). The first question is: which \(\sigma \)-field should we use? The power set \({\cal P}(\R )\) is too big, for reasons that we will make clear in Section 1.6. However, for practical purposes we do need our \(\sigma \)-field to contain all open and closed intervals, and also unions, intersections and complements of these. This provides a starting point.

Definition 1.4.2 The Borel \(\sigma \)-field of \(\R \), denoted by \({\cal B}(\R )\), is the smallest \(\sigma \)-field on \(\R \) that contains all open intervals \((a, b)\) where \(-\infty \leq a < b \leq \infty \). Sets in \({\cal B}(\R )\) are called Borel sets.

Note that \({\cal B}(\R )\) also contains isolated points \(\{a\}\) where \(a \in \R \). To see this first observe that \((a, \infty ) \in {\cal B}(\R )\) and also \((-\infty , a) \in {\cal B}(\R )\). Now by (S2), \((-\infty , a] = (a, \infty )^{c} \in {\cal B}(\R )\) and \([a, \infty ) = (-\infty , a)^{c} \in {\cal B}(\R )\). Finally as \(\sigma \)-fields are closed under intersections, \(\{a\} = [a, \infty ) \cap (-\infty , a] \in {\cal B}(\R )\). You can show that \({\cal B}(\R )\) also contains all closed intervals – see Problem 1.6. With open and closed intervals in hand, the closure of \(\sigma \)-fields under countable set operations gives us a way to construct a huge variety of Borel sets.

As a general rule, all ‘sensible’ subsets of \(\R \) are Borel sets. We might hope to find some sort of formula for a general element of \(\mc {B}(\R )\), but this is not possible. Unless you deliberately set out to find a non-Borel subset of \(\R \) you will never come across one – and even when you look for them it is hard work to find them, as we will see in Section 1.6.