Sigma Algebras

A \(\sigma\)-Algebra \(M\), of subsets of a set \(X\), is a collection of subsets of \(X\) that satisfies the following properties:

\begin{align} (1)&\ \text{M contains the empty set: } \emptyset \in M \\ (2)&\ \text{Closure under complementation: If } B \in M,\ \text{then } B^c \in M \\ (3)&\ \text{Closure under countable union: If } B_1, B_2, \ldots \in M,\ \text{then } \bigcup_{n=1}^{\infty} B_n \in M \end{align}

There are two things to consider with this definition. The first is that since \(\emptyset \in M\), and \(M\) is closed under complementation, it follows that \(\emptyset^c \in M\) and \(\emptyset^c = X \implies X \in M\).

The second concerns the choice of countable union. For some set \(E \in M\), where

E = \bigcup_{n=1}^{\infty}E_n

and where each \(E_n\) is some element of \(M\). Then we know \(E \in M \implies E^c \in M\). So if we replace \(E\) by the union of sets that it's comprised of, we get

E^c = \left(\bigcup_{n = 1}^{\infty}E_n\right)^c \] \[ (\bigcup_{n = 1}^{\infty}E_n)^c = \bigcap_{n = 1}^{\infty}E_n^c

Now, we know each \(E_n \in M \implies E_n^c \in M\). So in our above expression, we have the countable intersection of elements in \(M\). Finally, since this simply rewrites \(E^c\), we have:

E^c = \bigcap_{n = 1}^{\infty}E_n^c

Therefore, \(\bigcap_{n = 1}^{\infty}E_n^c \in M\). Combining closure under complementation and De Morgan's Law with closure under countable union, we conclude that \(M\) is also closed under countable intersection. Thus, we can equivalently define a \(\sigma\)-algebra using closure under countable intersection.

Why Are \(\sigma\)-Algebras Important?

In measure theory, for an exterior measure defined on a set \(X\), the measure assigns a value to every subset of \(X\) (the power set \(2^X\)). However, the exterior measure \(m_{\ast}\) is only countably sub-additive. That is,

\text{If } E = \bigcup_{j = 1}^{\infty}E_n \quad \text{then} \quad m_{\ast}(E) \leq \sum_{j = 1}^{\infty}m_{\ast}(E_n)

If we instead restrict the exterior measure to an appropriate \(\sigma\)-algebra of \(X\), then we actually get countable additivity:

\text{If } E = \bigcup_{j = 1}^{\infty}E_n \quad \text{then} \quad m_{\ast}(E) = \sum_{j = 1}^{\infty}m_{\ast}(E_n)\] \[\text{Where each } E_n \text{ is disjoint}

By "appropriate \(\sigma\)-algebra", I mean one that ensures countable additivity. For example, the power set \(2^X\) is a \(\sigma\)-algebra but does not allow for countable additivity. With the Lebesgue Exterior Measure, measurability carves out the right subsets:

\text{A set } \mathbb{E} \text{ is measurable if } \forall \epsilon > 0,\ \exists\ \mathbb{O} \supset \mathbb{E},\ \mathbb{O}\ \text{open, such that } m_{\ast}(\mathbb{O} - \mathbb{E}) \leq \epsilon \] \[\mathbb{O}-\mathbb{E} = \mathbb{O}\cap\mathbb{E}^c

Considering all subsets of \(X\) that satisfy this yields a \(\sigma\)-algebra that gives countable additivity.

More generally, \(\sigma\)-algebras provide structure to make conclusions about the behavior of set operations and how sets interact.

Another important example is the Borel \(\sigma\)-Algebra, defined as the smallest \(\sigma\)-algebra containing all open sets of \(\mathbb{R}^d\). Explicitly:

\text{A } G_{\delta} \text{ is a countable intersection of open sets}\] \[\text{An } F_{\sigma} \text{ is a countable union of closed sets}

The Borel \(\sigma\)-algebra adjoined with null sets forms the \(\sigma\)-algebra of Lebesgue Measurable Sets, the main objects of Lebesgue measure theory.