The Lebesgue Decomposition Theorem for Partially Ordered Semigroup-Valued Measures

Abstract

Abstract.The present paper is concerned with partially ordered semigroup-valued measures. 1. Preliminaries. By a partially ordered semigroup X we mean a commutative semigroup with identity 0, equipped by a partial ordering <, compatible with the structure of X under the conditions:(i) If x, y, z are elements of X with x < y (x < y and x ^ y) then x + z < y + z.(ii) x + sup E = sup (x + E), whenever there exist sup E (the supremum of E in X) and sup(x + E),E E X,x E X. Now X is monotone complete if every majorised increasing directed family in X has a supremum in X. Moreover, X is of the countable type if every subset £ of A" that has a supremum in X, contains a countable subset E* Ç E so that: sup E = sup E*.Let A" be a partially ordered semigroup and H a ring of subsets of T. The function m: H -» X is an o-measure (order measure) on H, if m is positive on H imiA) > 0, for every A in H) and mi\J"eNAn) = sup{2^_,w(^,): « E N} whenever iAn)neN is a disjoint sequence of elements of H with ( U "eAi^,) E H.The following propositions can be easily proved. Proposition 1.1. Let m: H -^ X be an o-measure on H.(1)«j(0) = 0.(2) m is finitely additive on H and miA) < miB), whenever A, B E H with A E B.(3) For every sequence iA")neN in H with (U"ejv^n) e H and sup{2ï_xmiAi): n E N) E X, implies: mi(JneNAn) < sup{2?=1m(v4,.): « 6 N).Received by the editors August 22, 1977 and, in revised form, November 30, 1977 AMS iMOS) subject classifications (1970). Primary 46G99; Secondary 28A55.Key words and phrases. Partially ordered semigroup, monotone complete partially ordered semigroup, partially ordered semigroup of the countable type, o-measure, absolutely continuous and singular o-measure, partially ordered topological semigroup, o-compatible topology with the partial ordering, T^-measure.

Decompositions of vector measures in Riesz spaces and Banach lattices

“…For the class of all order countably additive vector measures, the Lebesgue null-set decomposition theorem is essentially due to Pavlakos [8]; see also Congost Iglesias [2]. Their result is a consequence of Theorem 2.6 since obca(^, E) is a band in oba(^, E).…”