Samples, events, and random variables are the fundamental concepts in probability theory. In this chapter, we shall introduce the definitions of these concepts based on measure theory.

**Definition 11.1** (Sample Space, Sample Point, and Event). Let (Ω, , *P*) be a probability space (see Definition 2.12). The set Ω is called the *sample space*, a point in Ω is called a *sample point*, and a subset of is called an *event*.

**Definition 11.2** (Discrete Probability Space). Let Ω = {ω_{1}, ω_{2},…} be a finite or countably infinite set. Let *P* be a set function defined as

and

where *p*_{1}, *p*_{2},… are nonnegative numbers whose sum is 1. Then *P* is a probability measure on 2^{Ω}, which is the collection of all subsets of Ω. The probability space (Ω, 2^{Ω}, *P*) is called a *discrete probability space*.

**Definition 11.3** (Almost Surely). Let be a probability space. A statement about outcomes is said to be true almost surely (a.s.), or with probability 1, if the set ...

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