Chapter One

Probability Space

# 1.1 Introduction/Purpose of the Chapter

The most important object when working with probability is the proper definition of the space studied. Typically, one wants to obtain answers about real-life phenomena which do not have a predetermined outcome. For example, when playing a complex game a person may be wondering: What are my chances to win this game? Or, am I paying too much to play this game, and is there perhaps a different game I should rather play? A certain civil engineer wants to know what is the probability that a particular construction material will fail under a lot of stress. To be able to answer these and other questions, we need to make the transition from reality to a space describing what may happen and to create consistent laws on that space. This framework allows the creation of a mathematical model of the random phenomena. This model, should it be created in the proper (consistent) way, will allow the modeler to provide approximate answers to the relevant questions asked. Thus, the first and the most important step in creating consistent models is to define a probability space which is capable of answering the interesting questions that may be asked.

We denote with Ω the set that contains all the possible outcomes of a random experiment. The set Ω is often called sample space or universal sample space. For example, if one rolls a die, Ω = {1, 2, 3, 4, 5, 6}. The space Ω does not necessarily contain numbers but rather some representation ...