Sigma Field
Remember that a probability space is a triple consisting of a sample space, \(\sigma\) -field, and probability measure: \((\Omega, \mathcal{F}, \mathbb{P})\).
We can easily define a probability space when \(\Omega\) is finite or countable (see probability space), but we run into trouble, when the sample space has uncountably infinite elements. There's no way for us to assign a non-zero probability to each element and still have \(\mathbb{P}(\Omega) = 1\) (see sum of uncountably many non-negative numbers must have at most countably many non-zero numbers if the sum is to be finite). So instead, we compromise and only assign a probability to the 'nice' subsets of \(\Omega\). In general, we run into a lot of trouble when we try to define probability measures on all \(2^\Omega\) subsets, some of which can get pretty weird if \(\Omega = [0,1]\). But engineers don't care about measuring the probability of weird subsets. They're just concerned with the things we might encounter in ordinary life, e.g. intervals on the real number line, so for those purposes, only trying to measure the probability on a \(\sigma\) -field is good enough.
1. Definition
Given a sample space \(\Omega\), a \(\sigma\) -field is a collection \(\mathcal{F}\) of subsets of \(\Omega\) such that:
- \(\emptyset \in \mathcal{F}\)
- If \(A \in \mathcal{F}\), then \(A^C \in \mathcal{F}\)
- If \(A_i \in \mathcal{F}\) for \(i \in \mathbb{N}\), then \(\cup_{i=1}^{\infty} A_i \in \mathcal{F}\)
A set \(A\) that belongs to \(\mathcal{F}\) is called an event or an \(\mathcal{F}\) measurable set. The pair \((\Omega, \mathcal{F})\) is called a measurable space. In words, (3) is often said "\(\mathcal{F}\) is closed under countable union"
2. Remark
It turns out that if \(A_i \in \mathcal{F}\) for \(i \in \mathbb{N}\), then \(\cap_{i=1}^{\infty} A_i \in \mathcal{F}\).
2.1. proof
We use DeMorgan's law. For each \(A_i\), we know that \(A_i^C \in \mathcal{F}\) by (2). Then, by \(3\), \(\cup_{i=1}^{\infty} A_i^C \in \mathcal{F}\). Then by DeMorgan's law and (2), \(\cap_{i=1}^{\infty} A_i \in \mathcal{F}\).
3. Proposition 1
Consider a sample space \(\Omega\). Let \(S\) be an index set (possibly countably or uncountably infinite), and suppose that for every \(s\in S\), we have a \(\sigma\) -field \(\mathcal{F}_s\) of \(\Omega\). Then \(\mathcal{F} = \cap_{s\in S} \mathcal{F}_s\) is also a \(\sigma\) -field.
Let \(C\) be a collection of subsets of \(\Omega\). Often, we want to talk about the smallest \(\sigma\) -field that contains \(C\). Let \(\sigma(C)\) be the intersection of all \(\sigma\) -fields that contain \(C\). We say that \(C\) generates \(\sigma(C)\)