On Monogenic Operators and Measures

Abstract

Abstract. The notion of a monogenic operator between linear lattices, generalizing that of a monogenic measure, is introduced and investigated. The decomposition of an operator into its monogenic and antimonogenic parts is established. Products of monogenic measures are also considered.Introduction. We introduce the notion of a monogenic operator between linear lattices, i.e. of an order bounded linear operator which is determined in some sense by its restriction to a given linear sublattice. We prove that monogenic operators form a band (Theorem 1). This yields the decomposition of an arbitrary operator into its monogenic and antimonogenic parts. In a special case where operators are identified with real-valued measures on a a-field we get a generalization of a result of Johnson [4]. We compare our decomposition with some classical decompositions in measure theory (Examples 1 and 2). We also consider (not necessarily direct) products of monogenic measures (Theorem 2 and Example 3). As far as the linear-lattice-theoretical terminology is concerned we mostiy follow Jameson's

Multiplicative decomposition of probability measures

“…Note that, in general, B çt BX®B2 (cf. [3]). The following lemma shows that fin[0, P, ®P2]cBx®B2 for every p-measure P( on U( and therefore, unique extensibility of measures is a multiplicative property.…”

“…YetB¡ = {v G ca(U;): \v\ is 03-monogenic} and B -{p e ca(U, ®il2): \p\ is (SX ® 032-monogenic}. By [3], B and B¡ are bands, BX®B2 c B and, in general, Bx ®B2 jí B . However, we have…”