What Is a Sigma-Field?

There are many ideas from set theory that undergird probability. One such idea is that of a sigma-field. A sigma-field refers to the collection of subsets of a sample space that we should use in order to establish a mathematically formal definition of probability. The sets in the sigma-field constitute the events from our sample space.

Definition

The definition of a sigma-field requires that we have a sample space S along with a collection of subsets of S. This collection of subsets is a sigma-field if the following conditions are met:

• If the subset A is in the sigma-field, then so is its complement A^C.
• If A_n  are countably infinitely many subsets from the sigma-field, then both the intersection and union of all of these sets is also in the sigma-field.

Implications

The definition implies that two particular sets are a part of every sigma-field. Since both A and A^C are in the sigma-field, so is the intersection. This intersection is the empty set. Therefore the empty set is part of every sigma-field.

The sample space S must also be part of the sigma-field. The reason for this is that the union of A and A^C must be in the sigma-field. This union is the sample spaceS.

Reasoning

There are a couple of reasons why this particular collection of sets is useful. First, we will consider why both the set and its complement should be elements of the sigma-algebra. The complement in set theory is equivalent to negation. The elements in the complement of A are the elements in the universal set that are not elements of A. In this way, we ensure that if an event is part of the sample space, then that event not occurring is also considered an event in the sample space.

We also want the union and intersection of a collection of sets to be in the sigma-algebra because unions are useful to model the word “or.” The event that A or B occurs is represented by the union of A and B. Similarly, we use the intersection to represent the word “and.” The event that A and B occurs is represented by the intersection of the sets A and B.

It is impossible to physically intersect an infinite number of sets. However, we can think of doing this as a limit of finite processes. This is why we also include the intersection and union of countably many subsets. For many infinite sample spaces, we would need to form infinite unions and intersections.

Related Ideas

A concept that is related to a sigma-field is called a field of subsets. A field of subsets does not require that countably infinite unions and intersection be part of it. Instead, we only need to contain finite unions and intersections in a field of subsets.