Weakly tight functions, their Jordan type decomposition and total variation in effect algebras

Abstract

In the present paper, we have studied envelopes of a function m defined on a subfamily E (containing 0 and 1) of an effect algebra L. The notion of a weakly tight function is introduced and its relation to tight functions is investigated; examples and counterexamples are constructed for illustration. A Jordan type decomposition theorem for a locally bounded real-valued weakly tight function m defined on E is established. The notions of total variation |m| on the subfamily E and m-atoms on a sub-effect algebra E (along with a few examples of m-atoms for null-additive as well as non null-additive functions) are introduced and studied. Finally, it is proved for a real-valued additive function m on a sub-effect algebra E that, m is non-atomic if and only if its total variation |m| is non-atomic.

Atoms and Dobrakov submeasures in effect algebras

“…Notice that, several authors [13,31,32,35] have made their contributions to the theory of atoms in an effect algebra L where the definition of an atom involves only the structure of L, while here in this paper and also in [20,21], we have coined and studied the concept of an atom of a function on L.…”

“…The notion of Dobrakov submeasure in effect algebras introduced here (and therefore results in Section 3) involves a weak form of subadditivity [3,21] of the given mapping; new techniques based on [10,11,30] are used in proving the results. Notice that, several authors [13,31,32,35] have made their contributions to the theory of atoms in an effect algebra L where the definition of an atom involves only the structure of L, while here in this paper and also in [20,21], we have coined and studied the concept of an atom of a function on L.…”

“…In 1995, Pap [27,28] further introduced and studied atoms of null-additive set functions. Some important contributions to the study of the theory of atoms can be seen in [18][19][20][21][22]33]. Avallone and Basile [2] devoted their study in proving a Marinacci uniqueness theorem in the context of atomless measures.…”

The range of non-atomic measures on effect algebras

“…Let m be a real-valued function defined on an effect algebra L. Firstly, we shall recall the notion of an atom of a measure m defined on an effect algebra L, which has been studied in [24,25]. …”

“…[3]). In Section 4, using the notion of an atom of a real-valued measure m [24,25], we have showed that the range of a locally bounded real-valued σ-additive, non-atomic function m on a D-lattice L is an interval (−m − (1), m + (1)); characterizations of nonatomicity of m are established and used in obtaining this result (cf. [4]).…”