Probability and Statistical Inference. Robert Bartoszynski

Probability and Statistical Inference - Robert Bartoszynski


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alt="images"/> can be defined as follows: let images be a function such that images for all images and images. We will assume in addition that images is continuous and bounded, although those conditions can be greatly relaxed in general theory.

      (in this case, images is referred to as a density of images). The full justification of this construction lies beyond the scope of this book, but we will give the main points. First, the definition (2.12) is applicable for all intervals images of the form images, and so on. Then we can extend images to finite unions of disjoint intervals by additivity (the class of all such finite unions forms a field). We can easily check that such an extension is unique; that is,

equation

      does not depend on the way interval images is partitioned into the finite union of nonoverlapping intervals images. This provides an extension of images to the smallest field of sets containing all intervals. If we show that images defined this way is continuous on the empty set, then we can claim that there exists an extension of images to the smallest images‐field of sets containing all intervals.

      Now, the decreasing sequences of intervals converging to the empty set are built of two kinds of sequences: “shrinking open sets” and “escaping sets,” exemplified as

equation

      and

equation

      We have here images and images. In the first case, images, where images is a bound for function images. In the second case, images.

      Let us finally consider briefly the third interpretation of probability, namely as a degree of certainty, or belief, about the occurrence of an event. Most often, this probability is associated not so much with an event as with the truth of a proposition asserting the occurrence of this event.

      The material of this section assumes some degree of familiarity with the concept of expectation, formally defined only in later chapters. For the sake of completeness, in the simple form needed here, this concept is defined below. In the presentation, we follow more or less the historical development, refining gradually the conceptual structures introduced. The basic concept here is that of a lottery, defined by an event, say images, and two objects, say images and images. Such a lottery, written simply images, will mean that the participant (X) in the lottery receives object images if the event images occurs, and receives object images if the event images occurs.

      The second concept is that of expectation associated with the lottery images, defined as