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
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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