Chapter 2 - Measure Theory

Abstract

The concept of measure is an extension of the concept of length. The essential properties of the length of an interval are (1) a measure is always nonnegative (nonnegativity); (2) the measure of the union of a countable number of nonoverlapping sets equals the sum of their measures (countable additivity); (3) the measure of the difference of a set and a subset is equal to the difference of their measures (monotonicity); and (4) every set whose measure is not zero is uncountable. H. Lebesgue presented a mathematically rigorous description of the class of sets for which he defined a measure satisfying Borel's postulates. This measure is well known as Lebesgue measure and is the most important and useful concept of a measure found on the set of real numbers to date. The chapter discusses a general procedure for constructing measures by first defining the measure on a ring of subsets of an arbitrary abstract nonempty set, then obtaining a measure on a s-ring using an outer measure. The chapter discusses the application of this procedure to construct Lebesgue measure on a set of real numbers, R as a concrete example.