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Hung T. Nguyen (New Mexico State University, USA), Vladik Kreinovich (University of Texas at El Paso, USA) and Gang Xiang (University of Texas at El Paso, USA)

Source Title: Intelligent Data Analysis: Developing New Methodologies Through Pattern Discovery and Recovery

Copyright: © 2009
|Pages: 27
DOI: 10.4018/978-1-59904-982-3.ch002

Chapter Preview

Top**What is a random set? An intuitive meaning.** What is a random set? Crudely speaking, a random number means that we have different numbers with different probabilities; a random vector means that we have different vectors with different probabilities; and similarly, a random set means that we have different sets with different probabilities.

How can we describe this intuitive idea in precise terms? To provide such a formalization, let us recall how probabilities and random vectors are usually defined.

**How probabilities are usually defined.** To describe probabilities, in general, we must have a set of possible situations , and we must be able to describe the probability *P* of different properties of such situations. In mathematical terms, a *property* can be characterized by the *set* of all the situations which satisfy this property. Thus, we must assign to sets , the probability value .

According to the intuitive meaning of probability (e.g., as frequency), if we have two disjoint sets and , then we must have . Similarly, if we have countably many mutually disjoint sets , we must have .

A mapping which satisfies this property is called *-additive.*

It is known that even in the simplest situations, for example, when we randomly select a number from the interval , it is not possible to have a -additive function which would be defined on *all* subsets of . Thus, we must restrict ourselves to a class of subsets of . Since subsets represent properties, a restriction on subsets means restriction on properties. If we allow two properties and , then we should also be able to consider their logical combinations , , and – which in set terms correspond to union, intersection, and complement. Similarly, if we have a sequence of properties , then we should also allow properties and which correspond to countable union and intersection. Thus, the desired family should be closed under (countable) union, (countable) intersection, and complement. Such a family is called a *-algebra.*

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