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

“…We thus obtain the Yosida-Hewitt band decomposition theorem of Congost Iglesias [2] and a new Lebesgue band decomposition theorem. We also generalize the Lebesgue null-set decomposition theorem of Pavlakos [8] and Congost Iglesias [2] for vector measures in a super Dedekind complete Riesz space. These two Lebesgue decompositions usually differ from each other; for order countably additive vector measures they coincide if and only if the dimension of the Riesz space is equal to one.…”

“…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).…”