This note discusses some mathematical misunderstandings about Savage (1954). It is shown that in his model the probability measure cannot be countably additive, that the set of events must be a u-algebra and not just an algebra, that Savage did not characterize all atomless finitely additive probability measures, and that the state space in his model, while infinite, does not have to be uncount...

A control system is said to be nite if the Lie algebra generated by its vector elds is nite dimensional. Su cient conditions for such a system on a compact manifold to be controllable are stated in terms of its Lie algebra. The proofs make use of the Equivalence Theorem of [7] and of the existence of an invariant measure on certain compact homogeneous spaces.

We study the class of depth zero Boolean algebras, both from a classical viewpoint and an effective viewpoint. In particular, we provide an algebraic characterization, constructing an explicit measure for each depth zero Boolean algebra and demonstrating there are no others, and an effective characterization, providing a necessary and sufficient condition for a depth zero Boolean algebra of ran...

This note is devoted to the study of geometric properties and the relationships between a projective space and an exponential class, both naturally associated with the positive elements in a commutative Banach algebra. Even though the motivating problem consists of understanding the geometry of the class of densities with respect to a given measure, the formulation can be carried out in general...

The Hughston-Jozsa-Wootters (HJW) theorem entails that any finite ensemble compatible with a given density operator can be prepared from a fixed initial state by operations on a spacelike separated system. In this paper, we generalize the HJW theorem to the case of arbitrary measures on the state space of a von Neumann algebra with hyperfinite commutant. In doing so, we also show that every POV...

This paper considers states on the Weyl algebra of the canonical commutation relations over the phase space R. We show that a state is regular iff its classical limit is a countably additive Borel probability measure on R. It follows that one can “reduce” the state space of the Weyl algebra by altering the collection of quantum mechanical observables so that all states are ones whose classical ...

In this paper we investigate some hereditary properties of amenability modulo an ideal of Banach algebras. We show that if $(e_alpha)_alpha$ is a bounded approximate identity modulo I of a Banach algebra A and X is a neo-unital modulo I, then $(e_alpha)_alpha$ is a bounded approximate identity for X. Moreover we show that amenability modulo an ideal of a Banach algebra A can be only considered ...

Let $mathfrak{L}$ be the Virasoro-like algebra and $mathfrak{g}$ itsderived algebra, respectively. We investigate the structure of the Lie triplederivation algebra of $mathfrak{L}$ and $mathfrak{g}$. We provethat they are both isomorphic to $mathfrak{L}$, which provides twoexamples of invariance under triple derivation.

in this paper we investigate some hereditary properties of amenability modulo an ideal of banach algebras. we show that if (e) is a bounded approximate identity modulo i of a banach algebra a and x is a neo-unital modulo i, then (e) is a bounded approximate identity for x. moreover we show that amenability modulo an ideal of a banach algebra a can be only considered by the neo-unital modulo...

Conventions and Notations (1) The measure space (M;M; d ) is interpreted as having M as the underly set, M the sigma algebra on M , and d the mesaure on M: If M is a metric space then M is taken as the sigma algebra generated by all its metric-open sets. (2) For two measure spaces (M;M; d ) ; (M1;M1; d 1) and (M M1;M M1; d d 1) is their product measure space. All measurs in the paper are consid...