Robert Bartoszynski - Probability and Statistical Inference

Infinite operations on events play a very important role in the development of the theory, especially in determining limiting probabilities.

The definitions below will prepare the ground for the considerations in the following chapters. In Chapter 2, we will introduce probability as a number assigned to an event. Formally, we will be considering numerical functions defined on events, that is, on subsets of the sample space . As long as is finite or countably infinite, we can take the class of all subsets of as the domain of definition of probability. In case of infinite but not countable (e.g., where is an interval, the real line, or a plane), it may not be possible to define probability on the class of all subsets of . Although the explanation lies beyond the scope of this book, we will show how the difficulties can be avoided by suitable restriction of the class of subsets of that are taken as events. We begin with the concept of closure under some operation.

Definition 1.4.1We say that the class of subsets of is closed under a given operation if the sets resulting from performing this operation on elements of are also elements of .

Complementation , finite union , infinite union , limits of sequences , are few examples of such operations.

Let and let consist of all subsets of that are finite. is closed under finite unions and all intersections, finite or not. Indeed, if are finite sets, then is also finite. Similarly, if are finite, then , and hence is also finite. However, is not closed under complementation: if is finite ( ), then is not finite, and hence . On the other hand, if is the class of all subsets of that contain some fixed element, say 0, then is closed under all intersections and unions, but it is not closed under complementation.

The following concepts have an important role in the theory of probability.

Definition 1.4.2A nonempty class of subsets of that is closed under complementation and all finite operations (i.e., finite union, finite intersection) is called a field . If is closed under complementation and all countable operations, it is called a ‐ field . Finally, if is closed under monotone passage to the limit, 4 it is called a monotone class .

Let us observe that Definition 1.4.2 can be formulated in a more efficient way. For to be a field, it suffices to require that if , then and (or and ). Any of these two conditions implies (by induction and De Morgan's laws) the closure of under all finite operations. Consequently, for to be a ‐field, it suffices to require that whenever then and (or and ); this follows again from De Morgan's laws. 5