In order to compute probabilities, one must restrict themselves to collections of subsets of the arbitrary space known as -algebras. Due to the Banach-Tarski paradox, it turns out that assigning probability measures to any collection of sets without taking into consideration the set's cardinality will yield contradictions. Thus, a special class of sets must be adhered to in order to correctly define the notion of a probability measure.
This section will develop specific types of set structures in which we can compute probabilities.
-algebras are by far the most important set structure defined here as they are the building blocks for defining probability measures.
A collection, , of subsets of , is a -algebra if
1) is closed under complements: if
2) is closed under countable unions: if
A collection, , of subsets of , is an algebra if
1) is closed under complements: if
2) is closed under finite unions: if
Let be a collection of subsets of . We call the smallest -algebra that contains or is generated by as where each is a -algebra that contains .
Define as the set of all subsets of . Since , since and thus . Let , then each . Therefore, since . Thus, is a -algebra of subsets of .
I claim that based on the definitions of a -algebra, for any set on , we can always find a generating -algebra, , such that . Since is a -algebra such that is the set of all subsets of , every -algebra of subsets of is contained in . Therefore, we can always find a -algebra for an arbitrary set .
As it turns out, there are certain set structures that are a fair bit weaker than that of the -algebra but they are considerably useful when attempting to generate -algebras and prove particularly tricky results in probability.
We call a collection of subsets of a monotone class if the following hold:
1) is closed under increasing unions: if then
2) is closed under decreasing intersections: if then
We call a collection of subsets of a -system if is closed under finite intersection, i.e.
A collection is called a -system if
Let be a -system, then by the definition of a -system we can derive the following property:
2-1) If such that , then
Let be a -system. Let such that . By property 2 and 3 of -systems, since . Thus, since . Thus, by property 2 again, we see that . By DeMorgan's law, .
Now, assume 1 and 2-1. Since and and . So , which gives property 2 of .
Lemma: If is a -system and a -system, then is a -algebra.
Let be a -system and a -system. Then, and . So, we need only to show that is closed under countable unions. Let , where 's are not necessarily disjoint. Hence, we can form a partition of the 's as follows: Let , , , ... , . Now we have a disjoint partition of the 's and since is also a -system, is closed under intersection, so and therefore is a -algebra.
We now will state and prove a useful theorem in the construction of probability measures, called the - theorem.
Let be a -system and a -system containing . Then
Let be the smallest -system containing , then is a -system as the intersection of all classes of the same type preserves the properties of that class. Let be the smallest -algebra containing , then is closed under all properties of a -system, so is also a -system that contains . Thus, , since is one such collection of the intersection of all -systems that contain . Now we must show that . We do so by defining the following collection:
Let , that is, if , then is a -system. Let , and since , we have . Let , such that . Given , we have . Since , we have that , and thus , so 2-1 holds (which implies that 2 holds). Let , such that the 's are disjoint. Since , then each is disjoint and since , we have that , and thus and is a -system.
Now, if we assume that , , since is a -system, then . Since , and thus . Thus, and . By using the results from the definition on , we see that . Again, by the construction of , we see that is a -system and . Take and to arrive at either or and by symmetry of the construction, it must be the case that and is both a -system and a -system. Using the previous lemma, we have that is a -algebra that contains . Thus, and we see that .
We now provide a construction that is used to generate probabilities that are defined on an algebra, .
We call an outer probability measure if ,