Chapter Two

Probability Measure

# 2.1 Introduction/Purpose of the Chapter

The previous chapter presents the concept of measurable space. A measurable space is a couple with Ω a non-empty set and a σ-algebra. Such a measurable space is ready for a probability measure. This is a measure defined on the events in the σ-algebra , taking values in the unit interval [0, 1]. This setup then can be applied to any random phenomena, from simple ones—for example, rolling a dice or tossing a coin—to complex ones such as running a survey to ascertain whether or not each person in a sample supports the death penalty, or modeling a laboratory investigation to study the best amount of a certain chemical and its effects on the yield of a product.

The probability measures the size of the events. Any probability measure is defined by two properties: the probability of Ω must equal to 1 and the measure must be countably additive. What is the reason for these two properties? The first one implies that all possible outcomes of the experiment are accounted for in Ω—the universe of the experiment. The second property is very logical. It is natural to have P(A ∪ B) = P(A) + P(B) for two disjoint events A and ...