On a Review of the Lebesgue Integrals with Application to Probability Theory

There are two primary motivations for the study of Lebesgue measure theory and they are

a) It is desirable to measure the length of any subset of the real line

b) It is desirable to have a theory of the integral in which the syllogism

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Holds in greatest possible generality. It turns out that both these desidereta are too ambiguous. In fact (a) is impossible. In order to have a feasible and useful theory of measuring sets, we must restrict attention to a particular class of sets. As for (b), we can certainly construct a theory of the integral in which (I) is easy and natural. But there is no ”optimal” theory.

The Lebesgue integral addresses both of the above issues very nicely. We shall invest a few pages in this work to providing a brief introduction to the pertinent ideas. We will not be able to prove all the results, but can state them all precisely and provide some elucidating examples. In this work we will discuss the application of some of the important results of the Lebesgue measure to probability theory. Note that the notion of length that we shall develop here is called a ”measure” However, probability dates to the days of B. Pascal (1633-1662) and even before, when gamlers wanted to anticipate the results of certain bets. The subject did not develop space, and was fraught with paradoxes and conundrums. It was not until 1933, when A.N. Komogorov (1903-1987) realized that measure theory was the correct language for formulating probabilistic statements, that the subject could be set on rigorous footing.

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