Probability Theory — Algebras

A probability space $(\mathrm{\Omega },\mathrm{𝒜︀},\mathbb{P})$ consists of three elements.

The first element is the sample space $\mathrm{\Omega }$. It is the space of all possible events. The second element is the $𝜎$-algebra, $\mathrm{𝒜︀}$. This is a collection of combinations of events in the sample space. In a sense, it is a set of sets. The final element is a function $\mathbb{P}$, which assigns a value to every element in $\mathrm{𝒜︀}$. Each element of the probaility space deserves discussion. This article is dedicated to properties of the $𝜎$-algebra $\mathrm{𝒜︀}$.

Algebras are collections of sets of $\mathrm{\Omega }$ satisfying three properties:

• $\mathrm{\Omega }$ is a member of the algebra. That is, the realization of all possible events is contained in the algebra.
• The algebra is closed under compliments. The compliment (with respect to the set $\mathrm{\Omega }$) of any element in the algebra will yield another element in the algebra.
• The algebra is closed under finite unions.

The last property is significant, because it is what distinguishes any algebra on $\mathrm{\Omega }$ from a $𝜎$-algebra on $\mathrm{\Omega }$. An algebra is called a $𝜎$-algebra if it satisfies the additional property:

• The algebra is closed under countable unions.

This final restriction distinguishes $\mathrm{𝒜︀}$ from more general algebras that could be formed on $\mathrm{\Omega }$.

The collection $\mathrm{𝒜︀}$ must be a $𝜎$-algebra if the triple $(\mathrm{\Omega },\mathrm{𝒜︀},\mathbb{P})$ is to be a probability space. The following results about algebras apply to probability spaces:

algebras contain $\varnothing$
All algebras contain the entire sample space $\mathrm{\Omega }$. All algebras are also closed under compliment. Therefore, all algebras contain the compliment of the entire sample space, ${\mathrm{\Omega }}^{𝑐}$. This set is the empty set $\varnothing$.

algebras are closed under intersection
Take a finite union of elements of an algebra. The set formed by the union is in the contained in the algebra, because of closure under finite unions. Call this set $𝐵$. The compliment of this set ${𝐵}^{𝑐}$ is also contained in $\mathrm{𝒜︀}$, because of closure under compliments. By De Morgan’s law, the compliment of a union of sets (i.e. ${𝐵}^{𝑐}$) is equivalent to the intersection of the compliments of the sets. This shows that algebras are closed under finite intersections.

Since De Morgan’s law extends to countable unions, $𝜎$-algebras can be said to be closed under countably many intersections. Algebras can generally only be said to be closed under finite intersections.

algebras are closed under differences
Set differences can be stated in terms of intersections, and the result follows.