\(\sigma\)-Algebra

Parent: Probability

Definition

Sigma Algebra

Informally, a \(\sigma\)-algebra on a given sample space \(\Omega\) tells us all the ways we might define valid events. For instance, if we’re rolling 6-sided dice, we might want to be able to define events like “All sides are even” or “All sides are above 4.” These events don’t directly belong to the sample space but can be constructed from the sample space.

Formally, for a given set \(\Omega\), the set \(F(\Omega)\) is called a \(\sigma\)-algebra on \(\Omega\) if

1. \(\varnothing, \Omega\) both belong to the set i.e \(\Omega, \varnothing \in F(\omega)\)
2. \(F(\Omega)\) is closed under complementation: if \(\sigma \in F(\Omega)\), then \(\overline{\sigma} := \Omega \ \sigma \in F(\Omega)\)
3. \(F(\Omega)\) is closed under countable unions: if \(\sigma_1, \sigma_2, ... \in F(\Omega)\), then \(\cup_{i=1}^{\infty} \sigma_i \in F(\Omega)\)

For a given set \(\Omega\), we can construct \(\sigma\)-algebras of different complexities. The simplest is \(F(\Omega) := \{\varnothing, \Omega\}\), which we can quickly check is a valid \(\sigma\)-algebra. The algebra contains the original set and the empty set; the algebra is closed under complementation; the algebra is closed under countable unions. The most complicated is the power set of \(\Omega\) i.e. \(F(\Omega) := P(\Omega)\).

Limits on Valid \(\sigma\)-algebras

TODO (18.175 Lecture 1)

Borel Sigma-Algebras

The Borel \(\sigma\)-algebra \(B\) on a topological space \(\Omega\) is the smallest \(\sigma\)-algebra containing all open sets of \(\Omega\)

Technicalities and Terminologies

Algebras

We use the term \(\sigma\) algebra, but what is an algebra and where does the \(\sigma\) bit come from? An algebra \(\mathcal{A}\) is a set of sets closed under finite union and complementation. We say that a measure \(\mu\) on \(\mathcal{A}\) is \(\sigma\)-finite if there exists a countable set \(A_n \in \mathcal{A}\) such that \(\mu(A_n) < \infty\) and \(\union A_n = \Omega\); in english, this means that if a set containing the entire sample space has finite generalized volume, then the measure is \(\sigma\)-finite.

Semi-algebras

A semi-algebra is a collection \(\mathcal{S}\) of sets closed under intersection such that \(S \in \mathcal{S}\) implies that the complement of \(S\), \(S^C\), is a finite disjoint union of sets in \(\mathcal{S}\); this means that the complement of every element in the semi-algebra can be constructed by a finite union of the sets in \(\mathcal{S}\).

Lemma: If \(\mathcal{S}\) is a semi-algebra, then the set of finite disjoint unions of sets in \(\mathcal{S}\) is an algebra called the algebra generated by \mathcal{S}$$. Translation: If I have a collection of sets such that the complement of any set can be constructed from the union of the other sets, then this collection of sets (the semi-algebra) gives rise to an algebra.

\(\pi\) Systems and \(\lambda\) Systems

A collection of sets \(P\) is a __\(\pi\)-system\(if the collectiion is closed under intersection. A collection of sets\)L\(is a\)\lambda$$-system if

1. The sample space \(\Omega \in L\)
2. If \(A, B \in L\) and \(A \subset B\), then \(B - A \in L\)
3. If \(A_n \in L\) and \(A_n \uparrow A\), then \(A \in L\) (TODO)

Theorem: if P is a \(\pi\)-ssytem and L is a \(\lambda\)-system that contains \(P\), then \(\sigma(P) \subset L\), where \(\sigma(A)\) denotes the smallest \(\sigma\)-algebra containing A.

Caratheodory Extension Theorem

How do we actually construct a measure (any measure!) for any set a \(\sigma\)-algebra or a Borel \(\sigma\)-algebra.

Theorem: If \(\mu\) is a \(\sigma\)-finite measure on an algebra \(A\), then \(\mu\) has a unique extension to the \(\sigma\)-algebra generated by A.

English: If a collection of sets closed under complement and finite union (the algebra) contains a set such that the generalized volume (the measure) of that set is equal to the sample space and is finite (\(\sigma\)-finite), then there is a unique way of measuring the collection of finite disjoint unions of sets in the algebra (the \(\sigma\)-algebra). This tells us that if we want a measure on a \(\sigma\)-algebra, constructing a measure on the algebra that generates its is sufficient.