ABSTRACT
This chapter develops the theory of Lebesgue measure of subsets of ℝ following the original approach of Lebesgue using inner and outer measure. It illustrates the need for the concept of measure of a set and measurable function by considering an alternate approach to integration developed by Lebesgue in 1928. It then uses the fact that every open subset of ℝ can be expressed as a finite or countable union of disjoint open intervals to define the measure of open sets, and then of compact sets. It also illustrates why it is necessary to consider the concept of measure of a set by considering an alternate approach to integration. A very important concept in the study of measure theory involves the idea of a property being true for all x except for a set of measure zero. One of the main advantages of the Lebesgue theory of integration involves the interchange of limits.
