THE LEBESGUE DECOMPOSITION OF MEASURES ON ORTHOMODULAR POSETS By G. T. RUTTIMANN and CHR. SCHINDLER* [Received 14th October 1985] 1. Introduction THE study of measures on orthomodular posets has its origin in the quantum logic approach to quantum mechanics [2,5,6,12, 15] and as a mathematical branch it became known as non-commutative measure theory (e.g. see [17]). The purpose of this paper is to investigate the Lebesgue decomposition of positive measures into positive measures in this non-commutative setting. The geometrical aspect of this notion is emphasized. The main theorems in §3 (3.4 & 3.5) present a condition for subcones of the cone of positive measures on an orthocomplete orthomodular poset L under which the requirement of a positive Lebesgue decomposi- tion is equivalent to the poset L to be a Boolean lattice. These conditions are met by the cone obtained by restricting normal positive linear functionals on a /BW-algebra to the complete orthomodular lattice of idempotents in this algebra. As an application of the aforementioned result we obtain a measure-theoretic characterization of associative /BW-algebras amongst all /BJV-algebras. This is the main result of §5. Paragraph 4 is concerned with certain permanence properties of the positive Lebesgue decomposition, i.e. the behaviour of this decomposi- tion for selected sets of probability measures under the formation of direct products and direct sums. In particular it is shown that the collection of all probability measures of a finite constnictible orthomodu- lar poset has the positive Lebesgue decomposition property. 2. Prerequisites Let us begin with the definitions and basic facts which pertain to this paper. Let (L, =£, ') be an orthocomplemented poset, U L > 1, with 0 as the least and 1 as the greater element. A pair (p, q) of elements of L is said to be orthogonal, denoted by p 1 q, provided p^q'. A subset D of L is said to be orthogonal if each pair (p, q) of D with p =£ q is orthogonal. Clearly, a subset C of L has at most one supremum, resp. infimum. We denote the supremum of C, resp. infimum of C, if such exists, by * Research supported by Schweizerischer Nationalfonds/Fonds nationalsuisse under grant numbers 2.807-0.83, 2.200-0.84. Quart. J. Mafli. Oxford (2), 37 (1986), 321-345 © 1986 Oxford University Press