chapter 1, Probability Theory

Excerpt

1.1. Probability Space

Probability theory is concerned with measurements of random phenomena and the properties of such measurements. This opening section discusses the formulation of event structures, the axioms that need to be satisfied by a measurement to be a valid probability measure, and the basic set-theoretic properties of events and probability measures.

1.1.1. Events

At the outset we posit a set S, called the sample space, containing the possible experimental outcomes of interest. Mathematically, we simply postulate the existence of a set S to serve as a universe of discourse. Practically, all statements concerning the experiment must be framed in terms of elements in S and therefore S must be constrained relative to the experiment. Every physical outcome of the experiment should refer to a unique element of S. In effect, this practical constraint embodies two requirements: every physical outcome of the experiment must refer to some element in S and each physical outcome must refer to only one element in S. Elements of S are called outcomes.

Probability theory pertains to measures applied to subsets of a sample space. For mathematical reasons, subsets of interest must satisfy certain conditions. A collection E of subsets of a sample space S is called a σ-algebra if three conditions are satisfied:

(S1) S∈E.

(S2) If E∈E, then E^c∈E, where E^c is the complement of E.

(S3) If the countable (possibly finite) collection E₁,E₂,…∈E, then the union E₁∪E₂∪…∈E.

Subsets of S that are elements of E are called events. If S is finite, then we usually take the set of all subsets of S to be the σ-algebra of events. However, when S is infinite, the problem is more delicate. An in-depth discussion of σ-algebras properly belongs to a course on measure theory and here we will restrict ourselves to a few basic points.

Since S is an event and the complement of an event is an event, the null set Ø is an event.