1 - Fundamentals of Measure and Integration Theory

Abstract

This chapter focuses on the fundamentals of measure and integration theory. It is possible to develop a theory of measure with the countable additivity requirement replaced by the weaker condition of finite additivity. The disadvantage of doing this is that the resulting mathematical equipment is much less powerful. However, a convincing physical justification of countable additivity has yet to be given. If the probability P(A) of an event A is to represent the long run relative frequency of A in a sequence of performances of a random experiment, P must be a finitely additive set function; but only finitely many measurements can be made in a finite time interval, so countable additivity is not inevitable on physical grounds. At present, almost all applications of measure theory in mathematics use countable rather than finite additivity. In discussing the theory of integration, the basic concepts of Borel measurable functions are needed.