On the decomposition of measures with applications to continuum physics

Abstract

In this note we show that some concepts of modern measure theory may be relevant to the formulation of theories in continuum physics. In particular we observe that a large class of measure spaces possesses a decomposition property which is useful in applications. This decomposition property is obtained as a corollary to a theorem of Kelley [10] concerning the "Lebesgue decomposition property," which result is based upon the investigations of Segal [3]. The mathematical notation and definitions (following Kelley [10]), are presented first; then a Lemma and the Decomposition are given. The relationship between these results and physical theories is presented in a series of remarks at the end . As a preliminary we establish some useful definitions and notation. I Let be a set and let sg/a be a ring of subsets of M whose union is ~ . We further assume that d / ~ has the property: if {d~ } is a sequence of sets in d¢ "~ such that if ~¢. _ d E d¢ '~, for all n and some d in d /~ , then U . d . ~ d¢ 'a~. By a measure v in we mean a non-negative, finite valued, countably additive set function defined on ~ ' a with the following additional property: if {~ ' . } is a disjoint sequence in ~.¢/a such that g . v ( d . ) converges, then U . d . a jfa~.2 A subfamily q / i n ~ ' a is called a a-ideal in d¢ 'a~ if and only if it is a ring of sets, the intersection of a member of q /wi th a member o f ~ / a is a member of q/, and any member of~¢¢ '~ which is a countable union of sets in ~ is itself a set in q/. A set ~ _ ~ is (locally) measurable if and only if ~¢ n ~ e J / fa for all ~¢ in J/¢'~. We denote the class of measurable sets in ~ by ~//¢ and observe that J g is a or-algebra of subsets of ~ containing ~ ,a .3 I f~¢ and cg are in J r , we write ~¢Ro~ if and only i f c ~ ~ ¢ contains no set