Soft Comput (2008) 12:559–565 DOI 10.1007/s00500-007-0228-1 ORIGINAL PAPER Effect algebras with the subsequential interpolation property Anna Avallone · Paolo Vitolo Published online: 21 July 2007 © Springer-Verlag 2007 Abstract We prove Brooks–Jewett, Vitali–Hahn–Saks and Nikodym Boundedness theorems for modular measures on lattice-ordered effect algebras with the subsequential inter- polation property. Keywords Effect algebras · Subsequential interpolation property · Vitali–Hahn–Saks theorem · Brooks–Jewett theorem · Nikodym boundedness theorem 0 Introduction Some of the more important results of measure theory are Brooks–Jewett, Vitali–Hahn–Saks and Nikodym Bounded- ness theorems, which hold for measures on Boolean rings with the Subsequential Interpolation Property (see e.g. Weber 1986). In the last years many authors investigated the validity of these theorems for measures on more general structures than Boolean rings. We recall for example Barbieri (2001); Hamhalter (1997); Dvurecenskij (1996); D’Andrea and de Lucia (1991); de Lucia and Pap (1995a,b); Avallone (2006). In particular, in de Lucia and Pap (1995a,b) the authors proved Brooks–Jewett theorem for measures on effect algebras with the sequential completeness property, which is stronger than the subsequential interpolation property, and gives a char- acterization of the uniform boundedness of a set of func- Supported by G.N.A.M.P.A. A. Avallone (B ) · P. Vitolo Dipartimento di Matematica, Università della Basilicata, Contrada Macchia Romana, 85100 Potenza, Italy e-mail: anna.avallone@unibas.it P. Vitolo e-mail: paolo.vitolo@unibas.it tions on such structures. It is known (see D’Andrea et al. 1991) that Nikodym Boundedness theorem and Vitali–Hahn– Saks theorems fail for measures on effect algebras, but in Avallone (2006) it is proved that these theorems hold for modular measures on lattice-ordered effect algebras with the sequential completeness property. In this paper we prove that Brooks–Jewett, Vitali– Hahn–Saks and Nikodym Boundedness theorems also hold for modular measures on lattice-ordered effect algebras with the subsequential interpolation property. We recall that effect algebras have been introduced by Bennett and Foulis (1994) for modelling unsharp measure- ment in a quantum mechanical system. They are a general- ization of many structures which arise in quantum physics (see Beltrametti and Cassinelli 1981) and in mathematical economics (see Butnariu and Klement 1993; Epstein and Zhang 2001), in particular of orthomodular lattices in non- commutative measure theory and MV-algebras in fuzzy mea- sure theory. After 1994, there have been a great number of papers concerning effect algebras (see Dvurecenskij and Pulmannova 2000 for a bibliography). 1 Preliminaries An effect algebra ( L , ⊕, 0, 1) is a structure consisting of a set L , two special elements 0 and 1, and a partially defined binary operation ⊕ on L × L satisfying the following condi- tions for every a, b, c ∈ L : (1) If a ⊕b is defined, then b ⊕a is defined and a ⊕b = b ⊕a. (2) If b ⊕ c and a ⊕ (b ⊕ c) are defined, then a ⊕ b and (a ⊕ b) ⊕ care defined and a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ c. (3) For every a ∈ L , there exists a unique a ⊥ ∈ L such that a ⊕ a ⊥ is defined and a ⊕ a ⊥ = 1. (4) If a ⊕ 1 is defined, then a = 0. 123