Lyapunov decomposition of measures on effect algebras
by
Paolo Vitolo
University of Basilicata, Potenza, Italy
Coauthors: Anna Avallone

Effect algebras have been introduced by D.J. Foulis and M.K. Bennett in 1994 (see [B-F]) for modelling unsharp measurement in a quantum mechanical system. They are a generalization of many structures which arise in quantum physics (see [B-C]) and in Mathematical Economics (see [B-K] and [E-Z]), in particular of orthomodular lattices in non-commutative measure theory and MV-algebras in fuzzy measure theory. After 1994, there have been a great number of papers concerning effect algebras. We refer to [D-P] for a bibliography.

In 1974 Kluvanek and Knowles (see [K-K]) proved a decomposition theorem of a closed vector measure on a s-algebra into the sum of a Lyapunov measure (i.e. with convex range on every interval) and an "anti-Lyapunov" measure. This decomposition theorem was based on a characterization of Lyapunov vector measures on s-algebras given by Knowles in [K] in 1974. Such a characterization involves the integral map, which is not defined in effect algebras.

In 1991 De Lucia and Wright in [D-W] proved a characterization of closed Lyapunov measures on s-algebras by a different condition, namely non-injectivity (i.e. for every non-negligible a, there exist b, c ≤ a with b\trianglec non-negligible and m(b)=m(c)). This result of De Lucia and Wright has been extended in 2003 by Avallone and Barbieri in [A-B] to modular measures on D-lattices, by replacing non-injectivity with another condition, called pseudo non-injectivity, which is equivalent to non-injectivity for measures on s-algebras. Precisely m is pseudo non-injective if for every non-negligible a, there exist orthogonal non-negligible elements b, c with b⊕c ≤ a and m(b)=m(c).

Starting from the result of Avallone and Barbieri, in this paper we prove the following Kluvanek-Knowles type decomposition theorem: Every closed modular measure on a s-complete D-lattice (i.e. a lattice ordered effect algebra) can be decomposed into the sum of a Lyapunov modular measure and an "anti-Lyapunov" modular measure.

[A-B] A. Avallone-G. Barbieri Lyapunov measures on effect algebras Comment. Math. Univ. Carolinae 44, no. 3, 387-397 2003

[B-C] E.G. Beltrametti-G. Cassinelli The logic of quantum mechanics Addison-Wesley Publishing Co., Reading, Mass. (1981)

[B-F] M.K. Bennett-D.J. Foulis Effect algebras and unsharp quantum logics Found. Phys. 24, no. 10, 1331-1352 1994

[B-K] D. Butnariu-P. Klement Triangular norm-based measures and games with fuzzy coalitions Kluwer Acad. Publ. (1993)

[D-P] A. Dvurecenskij-S. Pulmannová New trends in quantum structures Kluwer Acad. Publ. (2000)

[D-W] P. de Lucia-J.M. Wright Group valued measures with the Lyapunoff property Rend. Circ. Mat. Palermo 40, no. 3, 442-452 1991

[E-Z] L.G. Epstein-J. Zhang Subjective probabilities on subjectively unambiguous events Econometrica 69, no. 2, 265-306 2001

[K-K] I. Kluvánek-G. Knowles Liapunov decomposition of a vector measure Math. Ann. 210, 123-127 1974

[K] G. Knowles Liapunov vector measures SIAM J. Control 13, no. 2, 294-303 1975

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Date received: April 21, 2008