The basic ingredients of the consistent histories approach to quantum mechanics are the space of histories and the space of decoherence functionals. In this work we extend the classification theorem for decoherence functionals proven by Isham, Linden and Schreckenberg to the case where the space of histories is the lattice of projection operators on an arbitrary separable or non-separable compl...

A space X is cometrizable if X has a coarser metric topology such that each point of X has a neighborhood base of metric closed sets. Most examples in the literature of spaces obtained by modifying the topology of the plane or some other metric space are cometrizable. Assuming the Proper Forcing Axiom (PFA) we show that the following statements are equivalent for a cometrizable space X : (a) X ...

We study the separability of the state space of loop quantum gravity. In the standard construction, the kinematical Hilbert space of the diffeomorphism-invariant states is nonseparable. This is a consequence of the fact that the knot-space of the equivalence classes of graphs under diffeomorphisms is noncountable. However, the continuous moduli labeling these classes do not appear to affect the...

Statistical modeling of space-time data has often been based on separable covariance functions, that is, covariances that can be written as a product of a purely spatial covariance and a purely temporal covariance. The main reason is that the structure of separable covariances dramatically reduces the number of parameters in the covariance matrix and thus facilitates computational procedures fo...

Some properties of Borel measures with separable supports are considered. In particular, it is proved that any σ-finite Borel measure on a Suslin line has a separable supports and from this fact it is deduced, using the continuum hypothesis, that any Suslin line contains a Luzin subspace with the cardinality of the continuum. Let E be a topological space. We say that the space E has the propert...

Throughout the sequel, E denotes a reflexive real Banach space and E∗ its topological dual. We also assume that E is locally uniformly convex. This means that for each x ∈ E, with ‖x‖ = 1, and each > 0, there exists δ > 0 such that, for every y ∈ E satisfying ‖y‖ = 1 and ‖x− y‖ ≥ , one has ‖x + y‖ ≤ 2(1 − δ). Recall that any reflexive Banach space admits an equivalent norm with which it is loca...

Making an extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or −1 or Ω) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorph...

we prove that the limit of a sequence of pettis integrable bounded scalarly measurable weak random elements, of finite weak norm, with values in the dual of a non-separable banach space is pettis integrable. then we provide basic properties for the pettis conditional expectation, and prove that it is continuous. calculus of pettis conditional expectations in general is very different from the c...

For a contingent differential equation that takes values in the closed, convex, nonempty subsets of a Banach space E, we prove an existence theorem and we investigate the extendability of solutions and the closedness and continuity properties of solution funnels. We consider first a space E that is separable and reflexive and then a space E with a separable second dual space. We also consider t...

A geometrical picture of separability of 2× 2 composite quantum systems, showing the region of separable density matrices in the space of hermitian matrices, is given. It rests on the criterion of separability given by Peres [1], and it is an extension of the “Horodecki diagram” [2] and the “stella octangula” described by Aravind [3]. An illustration of which composite quantum states that are s...