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IntroductionThe well-known spectral theorem for self-adjoint operators on a Hilbert space can be formulated as follows:Let H be a complex separable Hilbert space with dim H > 2 and let L(H) denote the orthomodular lattice (shortly, OML) of all orthogonal projections from H onto closed linear subspaces of H. Let O denote the set of all self-adjoint linear operators on H and {m Q \ a (E S} the set of all pure probability measures on L(H). Then for every A 6 O there exists a unique Z(//)-valued measure (spectral measure) HA on B(R) such that for every a E S the composed mapping m a o is a probability measure on B(R). (Here and in the following B(R) denotes the Boolean a-algebra of all Borel sets of the real line.) Hence, the spectral theorem determines a doubly indexed family (PA,a)Aeo,aes of probability measures on B(R) such that each Pa,o can be decomposed in the form p A ,a = f^a 0 Pa where /i^ is an £(#)-valued measure on B(R) and m a is a pure probability measure on L(H). The family (PA,a)Aeo,aes can be interpreted as the spectral family of probability measures on B(R) corresponding to O and S. Now, by the inverse spectral theorem we may understand the following problem:Given a doubly indexed family (PA,a)Aeo,aes of probability measures on B(R), what conditions are to be put on O and S in order that every This paper is a result of the collaboration of the three authors within the framework of the Partnership Agreement between the University of Technology Vienna and the Warsaw University of Technology. The authors are grateful to both universities for providing financial support which has made this collaboration possible.

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Let S be a set of states of a physical system. The probabilities p(s) of the occurrence of an event when the system is in different states s ∈ S define a function from S to [0, 1] called a numerical event or, more precisely, an S-probability. If one orders a set P of S-probabilities in respect to the order of functions, further includes the constant functions 0 and 1 and defines p = 1 − p for every p ∈ P , then one obtains a bounded poset of S-probabilities with an antitone involution. We study these posets in respect to various conditions about the existence of the sum of certain functions within the posets and derive properties from these conditions. In particular, questions of relations between different classes of S-probabilities arising this way are settled, algebraic representations are provided and the property that two S-probabilities commute is characterized which is essential for recognizing a classical physical system.