Measurable Sets

This chapter defines the measurable subsets of R d that constitute the second step in the construction of the integral, over such a set, of an integrable measure. It presents the properties of the measure of open sets that serve to define, measurable sets and their measure, which is the Lebesgue measure. The chapter shows that measurability is preserved under finite or countable unions and intersections. It also shows that the measure is monotonic, that it is additive over finite or countable unions of disjoint sets, and that it is continuous with respect to increasing or decreasing sequences of sets. The chapter shows that any countable union and any countable intersection of measurable sets are measurable. It also defines negligible sets, which are of interest because the integral is not affected by the inclusion of such a set into the domain of integration.