Decomposition and representation theorems in measure theory

Abstract

"= the class of sets of finite measure and, as far as integration theory is concerned, this is no real restriction. A subfamily ~ of d is a a.ideal in d iff ~ is a ring of sets, the intersection of each member of ~ with a member of d is a member of ~ and each member of ~¢ which is the countable union of members of itself belongs to ~. A subset E of X is called measurable iff A ~ E C d for each A in d. (Some authors call such sets locally measurable). The class of measurable sets is a a-algebra of subsets of X. If E and F are measurable subsets of X then we agree that E ~_~ F Life ~ E contains no set of positive measure. If E and F are members of ~¢ then E _~ ~ P iff m (F ~ E) = 0. Clearly ~ ~ is a partial ordering of the class of measurable sets. We further agree that E is m.~uivalen~ to F,