Measure Theory

Abstract

If topology derives its inspiration from the qualitative features of geometry, then the subject of the present chapter, measure theory, may be thought to have its origins in the quantitative concepts of length, area, and volume. However, a careful theory of area, for example, turns out to be much more delicate than one might expect initially, as any given set may possess an irregular feature, such as having a jagged boundary or being dispersed across many subsets. Even in the setting of the real line, if one has a set E of real numbers, then in what sense can the length of the set E be defined and computed? Furthermore, to what extent can we expect the length (or area, volume) of a union A ∪ B of disjoint sets A and B to be the sum of the individual lengths (or areas, volumes) of A and B?