Probability with Measure

1.2 Sigma Fields

So far we have only discussed length but now we want to be more ambitious. Let \(S\) be an arbitrary set. We want to define mappings from subsets of \(S\) to \([0, \infty ]\) which we will continue to denote by \(m\). These will be called measures and they will share some of the properties that we’ve just been looking at for measures of length. Now on what type of subset of \(S\) can \(m\) be defined? The power set of \(S\) is \({\cal P}(S)\) and we have just argued that this could be too large for our purposes as it may contain sets that can’t be measured.

Suppose that \(A\) and \(B\) are subsets of \(S\) that we can measure. Then we should surely be able to measure the complement \(A^{c}\), the union \(A \cup B\) and the whole set \(S\). Note that we can then also measure \(A \cap B = (A^{c} \cup B^{c})^{c}\). This leads to a definition

  • Definition 1.2.1 Let \(S\) be a set. A Boolean algebra \(\mathbf B\) is a set of subsets of \(S\) that has the following properties

    • B(i) \(S \in {\mathbf B}\),

    • B(ii) If \(A,B \in {\mathbf B}\) then \(A \cup B \in {\mathbf B}\),

    • B(iii) If \(A \in {\mathbf B}\) then \(A^{c} \in {\mathbf B}\).

Note that \(\mathbf {B}\) is a set, and each element of \(\mathbf {B}\) is a subset of \(S\). In other words, \(\mathbf {B}\) is a subset of the power set \(\mc {P}(S)\). In this course we will frequently work with sets, whose elements are sets. It’s important to get used to working with these objects; don’t forget the difference between \(\{\{1\},\{2\}\}\) and \(\{1,2\}\).

Boolean algebras are named after the British mathematician George Boole (1815-1864) who introduced them in his book The Laws of Thought published in 1854. They are well studied mathematical objects that are extremely useful in logic and digital electronics. It turns out that they are inadequate for our own purposes – we need a little more sophistication.

If we use induction on B(ii) then we can show that, if \(A_{1}, A_{2}, \ldots A_{n} \in {\mathbf B}\) then \(A_{1} \cup A_{2} \cup \cdots \cup A_{n} \in {\mathbf B}\). This is left for you to prove, in Problem 1.1. But we need to be able to do analysis and this requires us to be able to handle infinite unions. The next definition gives us what we need:

  • Definition 1.2.2 Let \(S\) be a set. A \(\sigma \)-field \(\Sigma \) is a set of subsets of \(S\) that has the following properties

    • S(i) \(S \in \Sigma \),

    • S(ii) If \((A_{n})\) is a sequence of sets with \(A_{n} \in \Sigma \) for all \(\nN \) then \(\bigcup _{n=1}^{\infty }A_{n} \in \Sigma \),

    • S(iii) If \(A \in \Sigma \) then \(A^{c} \in \Sigma \).

The terms \(\sigma \)-field and \(\sigma \)-algebra have the same meaning (this is an unfortunate accident of history!). Often you will find that ‘\(\sigma \)-field’ is used in advanced texts and ‘\(\sigma \)-algebra’ is used within lecture courses. I prefer \(\sigma \)-field, you may use either.

Lastly, a piece of terminology.

  • Definition 1.2.3 Given a \(\sigma \)-field \(\Sigma \) on \(S\), we say that a set \(A \subset S\) is measurable if \(A \in \Sigma \)

Facts about \(\sigma \)-fields

  • • By S(i) and S(iii), \(\emptyset = S^{c} \in \Sigma \).

  • • We have seen in S(ii) that infinite unions of sets in \(\Sigma \) are themselves in \(\Sigma \). The same is true of finite unions. To see this let \(A_{1}, \ldots , A_{m} \in \Sigma \) and define the sequence \((A_{n}^{\prime })\) by \(A_{n}^{\prime } = \left \{ \begin {array}{c c} & A_{n} ~\mbox {if}~1 \leq n \leq m\\ & \emptyset ~\mbox {if}~n > m \end {array} \right .\) Now apply S(ii) to get the result. We can deduce from this that every \(\sigma \)-field is a Boolean algebra.

  • • \(\Sigma \) is also closed under infinite (or finite) intersections. To see this use de Morgan’s law to write

    \[ \bigcap _{i=1}^{\infty } A_{i} = \left (\bigcup _{i=1}^{\infty } A_{i}^{c}\right )^{c}.\]

  • • \(\Sigma \) is closed under set theoretic differences \(A-B\), since (by definition) \(A-B = A \cap B^{c}\).

Examples of \(\sigma \)-fields

  • 1. \({\cal P}(S)\) is a \(\sigma \)-field. If \(S\) is finite with \(n\) elements then \({\cal P}(S)\) has \(2^{n}\) elements (Problem 1.2).

  • 2. For any set \(S\), \(\{\emptyset , S\}\) is a \(\sigma \)-field which is called the trivial \(\sigma \)-field. It is the basic tool for modelling logic circuitry where \(\emptyset \) corresponds to “OFF” and \(S\) to “ON”.

  • 3. If \(S\) is any set and \(A \subset S\) then \(\{\emptyset , A, A^{c},S\}\) is a \(\sigma \)-field .

  • 4. The most important \(\sigma \)-field for studying the measure of length is the Borel \(\sigma \)-field of \(\R \) which is denoted \({\cal B}(\R )\). It is named after the French mathematican Emile Borel (1871-1956) who was one of the founders of measure theory. It is defined rather indirectly and we postpone this definition until after the next section.

A pair \((S, \Sigma )\) where \(S\) is a set and \(\Sigma \) is a \(\sigma \)-field of subsets of \(S\) is called a measurable space There are typically many possible choices of \(\Sigma \) to attach to \(S\). For example we can always take \(\Sigma \) to be trivial or the power set. The choice of \(\Sigma \) is determined by what we want to measure.