Generating σ-fields

For a sample space $\Omega$, let $\cA$ be a set of subsets of $\Omega$. Suppose we want to generate a $\sigma$-field using $\cA$. Note that there may be many different $\sigma$-fields $\cF$ such that $\cA\subseteq\cF$. Hence, to make this generation unique, we will focus on the smallest such $\cF$.

Definition: σ-field generated by A

The $\sigma$-field generated by $\cA$, denoted by $\sigma(\cA)$, is the intersection of all the $\sigma$-fields $\cF_i$ such that $\cA\subseteq\cF_i$. Therefore, it is the smallest $\sigma$-field containing $\cA$.

Examples

1. For $\cA=\{\emptyset\}$ or $\cA=\{\Omega\}$, $\sigma(\cA)=\{\emptyset, \Omega\}$, the trivial $\sigma$-field.
2. For $\cA=\{A\}$ for some $A\subset\Omega$, $\sigma(\cA)=\{\emptyset, A, A^c, \Omega\}$.
3. For $\cA=\{\{\omega\}: \omega\in\Omega\}$, $\sigma(\cA)=\{A: A$ is countable or $A^c$ is countable $\}$.

Exercise 2

1. Show that for any collection $\cA$ of subsets of $\Omega$, $\sigma(\cA)$ is indeed a $\sigma$-field.
2. Verify example 3 above.

The Borel σ-field

Denoted by $\cB$, the Borel $\sigma$-field is defined on the sample space $\Omega=(0,1]$. It is the $\sigma$-field generated by $\cB_0 = \{\text{finite unions of disjoint subintervals of }\Omega\}$ (recall that we saw this set earlier as a counterexample for $\sigma$-fields). Note that one can define the Borel $\sigma$-field on $\R$ in the same way as we did for $\Omega$.

It is well known that $\cB$ can also be generated by open, closed, and half-open sets. Formally, let

• $\cQ=\{(a, b]\subseteq\Omega\}$
• $\cR=\{(a, b)\subseteq\Omega\}$
• $\cS=\{[a, b]\subseteq\Omega\}$
• $\cT=\{[a, b)\subseteq\Omega\}$

The notation above may feel a little weird, but essentially it says that $\cQ$ is the set of all intervals of $\Omega$ that are open on the left and closed on the right (similarly for the other three sets).

Exercise 3

Problem

Prove that $\sigma(\cQ) = \sigma(\cR) = \sigma(\cS) = \sigma(\cT) = \cB$.

Hint

The countably infinite union of a carefully chosen sequence of left-open intervals (elements of $\cQ$) can generate an open interval (an element of $\cR$).

Similarly, the countably infinite intersection of a chosen sequence of open intervals (elements of $\cR$) can generate a closed interval (an element of $\cS$), and so on.