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Sigma Algebras

Rings, Algebras, \(\sigma\)-algebras of sets#

Definition. A nonempty family of sets \(\mathcal{R}\) on a nonempty set \(X\) is called a ring of sets if it is closed under finite unions and relative complement (\(A,B\in \mathcal{R}\) implies \(A\setminus \in \mathcal{R}\)). An algebra of sets is a ring of sets together with \(X\), that is, it is a family of sets closed under finite unions and complements.

Definition. A \(\sigma\)-algebra is an algebra that is closed under countable unions.

Since \(\cap_j E_j = (\cup_j E^c_j)^c\) algebras (and \(\sigma\)-algebras) are also closed under finite (and countable) intersections. For an algebra, both \(\emptyset\) and \(X\) belongs to it since \(E\cap E^c = \emptyset\) and \(E\cup E^c =X\). It is also worth nothing that an algebra is a \(\sigma\)-algebra provided that it is closed under countable disjoint unions.

Since the intersection of any family of \(\sigma\)-algebras on \(X\) is again a \(\sigma\)-algebra on it, it follows that given a subset \(\mathcal{E}\) of \(\mathcal{P}(X)\), there is a unique smallest \(\sigma\)-algebra \(\mathcal{M}(\mathcal{E})\) containing \(\mathcal{E}\), namely the intersection of all \(\sigma\)-algebras containing \(\mathcal{E}\); there is always at least one such, namely \(\mathcal{P}(X)\). \(\mathcal{M}(\mathcal{E})\) is called the \(\sigma\)-algebra generated by \(\mathcal{E}\).

It is easy to see that if \(\mathcal{E}\subset\mathcal{M}(\mathcal{F})\) then \(\mathcal{M}(\mathcal{E}) \subset \mathcal{M}(\mathcal{F})\).

Elementary family#

An elementary family is a collection \(\mathcal{E}\) of subsets of \(X\) such that 

  • \(\empty\in\mathcal{E}\);
  • \(E,F\in\mathcal{E}\) then \(E\cap F\in\mathcal{E}\);
  • \(E\in\mathcal{E}\) then \(E^c\) is a finite disjoint union of members of \(\mathcal{E}\).

Proposition. If \(\mathcal{E}\) is an elementary family, then the collection \(\mathcal{A}\) of finite disjoint unions of members of \(\mathcal{E}\) is an algebra.

Examples#

Borel \(\sigma\)-algebra#

If \(X\) is a topological space, then the family of open sets in \(X\) generates a \(\sigma\)-algebra, called the Borel \(\sigma\)-algebra on \(X\), the members of which are called Borel sets. We denote the Borel \(\sigma\)-algebra by \(\mathcal{B}_X\). It incluides open sets, closed sets, countable intersections of open sets, countable unions of closed sets, etc. Recall that there is a hierarchy of sets, The Boldface Borel Hierarchy of finite rank.

In particular, the \(\mathcal{B}_\mathbb{R}\) plays a fundamental role in measure theory. It is easy to see that is can be generated in a number of different ways, by 

  • the open intervals (these are just countable unions of open sets)
  • the half-open intervals (an open interval is just the intersection of two half-open intervals )
  • the open rays (same with the half-open intervals)
  • the closed rays (complement of the open rays)
  • the closed intervals (intersections of closed rays)

Product \(\sigma\)-algebra#

Let \(\{X_\alpha\}_{\alpha\in A}\) be an index collection of nonempty sets, \(X = \prod_{\alpha \in A} X_\alpha\), and \(\pi_\alpha: X\to X_\alpha\) the coordinate maps. If \(\mathcal{M}_\alpha\) is a \(\sigma\)-algebra on \(X_\alpha\) for each \(\alpha\), the product \(\sigma\)-algebra on \(X\), denoted \(\otimes_{\alpha\in A}\mathcal{M}_\alpha\), is the \(\sigma\)-algebra generated by

\[\{\pi^{-1}_\alpha(E_\alpha):E_\alpha\in\mathcal{M}_\alpha,\alpha\in A\}.\]

If \(A\) is countable, \(\otimes_{\alpha\in A}\mathcal{M}_\alpha\) is the \(\sigma\)-algebra generated by

\[\{\prod_{\alpha\in A}E_\alpha:E_\alpha\in\mathcal{M}_\alpha\}.\]