ABSTRACT
This chapter is devoted to a quantitative theory of possibility. We will use some facts about random sets to provide a firm foundation for possibility measures.
As discussed earlier, uncertainty arises in real-world problems in various forms, of which randomness is only one. In formulating the structure of a random experiment, such as games of chance, we list the collection of all possible outcomes of the experiment. It is on the concept of chance that we focus our attention. The theory of probability is first of all intended to provide an acceptable concept of quantitative chance. Despite the fact that events can be possible or probable, the lack of a quantitative theory of possibility has led us to focus only on probability. One of the contributions of probability theory to science is the rigorous quantification of the concept of chance, together with a solid theory of this quantification. This chapter is focused on the quantification of another type of uncertainty, namely possibility. But it is appropriate to review basic facts about probability theory, especially the concept of random sets, which will play an essential role in developing the foundation for other uncertainty measures, including belief functions in Chapter 10. The quantitative concept of chance arose from games of chance, and
later was formulated in general contexts via an axiomatic approach due to Kolmogorov. The uncertainty measure involved is a set function known as a probability measure. Specifically, the mathematical model of a random experiment is this. A set Ω is intended to represent the set of all possible outcomes of the experiment, called the sample space. This name
consists of choosing, in some random fashion, a sample from the population. A collection A of subsets of Ω is intended to represent events whose probabilities can be assigned or defined. Here is the formal definition of the concepts.
