last updated: May 9, 2024

Probability with Measure

1.4 The Borel σ-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 ba, 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 I and the rational numbers Q. Both 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 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 σ-field should we use? The power set P(R) is too big, for reasons that we will make clear in Section 1.6. However, for practical purposes we do need our σ-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 σ-field of R, denoted by B(R), is the smallest σ-field on R that contains all open intervals (a,b) where a<b. Sets in B(R) are called Borel sets.

Note that B(R) also contains isolated points {a} where aR. To see this first observe that (a,)B(R) and also (,a)B(R). Now by (S2), (,a]=(a,)cB(R) and [a,)=(,a)cB(R). Finally as σ-fields are closed under intersections, {a}=[a,)(,a]B(R). You can show that B(R) also contains all closed intervals – see Problem 1.6. With open and closed intervals in hand, the closure of σ-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 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.