Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

CHAPTER 11

EVENTS AND RANDOM VARIABLES

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.

11.1 Basic Concepts and Facts

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 = {ω₁, ω₂,…} be a finite or countably infinite set. Let P be a set function defined as

and

where p₁, p₂,… 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 ...