Related papers: Tarski Measure
We examine measure-theoretic properties of spaces constructed using certain technique of Todor\v{c}evi\'{c}. We show that the existence of strictly positive measures on such spaces depends on combinatorial properties of certain families of…
We introduced the concept of a metric value set (MVS) in an earlier paper \cite{GM} and developed the idea further in \cite{AS}. In this paper we study locally $M$-metrizable spaces and the products of $M$-metrizable spaces. Finally we…
A marked metric measure space (mmm-space) is a triple (X,r,mu), where (X,r) is a complete and separable metric space and mu is a probability measure on XxI for some Polish space I of possible marks. We study the space of all (equivalence…
We introduce the notion of scale to generalize and compare different invariants of metric spaces and their measures. Several versions of scales are introduced such as Hausdorff, packing, box, local and quantization. They moreover are…
It is shown that any transverse invariant measure of a foliated space can be considered as a measure on the ambient space.
We propose a definition of magnitude for a length space with a Borel measure, which involves integrals over the set of geodesics. This quantity agrees with the magnitude of finite metric spaces, up to re-scaling the metric to ensure the…
A "tensor space" is a vector space equipped with a finite collection of multi-linear forms. In previous work, we showed that (for each signature) there exists a universal homogeneous tensor space, which is unique up to isomorphism. Here we…
The hermitian analog of Aleksandrov's area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area…
Using the recently defined concept of Taylor measures, we propose a generalization of Taylor's theorem to measurable, non-analytic functions, that do not require differentiation. We study consequences of the generalization, including the…
Using the kinematic constraints of classical bodies we construct the allowable wavefunctions corresponding to classical solids. These are shown to be long lived metastable states that are qualitatively far from eigenstates of the true…
Gaussian processes can be considered as subsets of a standard Hilbert space, but the geometric understanding that would relate the size of a set with the size of its convex hull is still lacking. In this work, we adopt a geometric approach…
Magnitude is a numerical isometric invariant of metric spaces, whose definition arises from a precise analogy between categories and metric spaces. Despite this exotic provenance, magnitude turns out to encode many invariants from integral…
Measurements play a crucial role in doing physics: Their results provide the basis on which we adopt or reject physical theories. In this note, we examine the effect of subjecting measurements themselves to our experience. We require that…
The celebrated Takens' embedding theorem provides a theoretical foundation for reconstructing the full state of a dynamical system from partial observations. However, the classical theorem assumes that the underlying system is deterministic…
In this manuscript, a conformally invariant theory of gravitation in the context of metric measure space is studied. The proposed action is invariant under both diffeomorphism and conformal transformations. Using the variational method, a…
We consider a randomly forced Ginzburg-Landau equation on an unbounded domain. The forcing is smooth and homogeneous in space and white noise in time. We prove existence and smoothness of solutions, existence of an invariant measure for the…
The theory of measurement is employed to elucidate the physical basis of general relativity. For measurements involving phenomena with intrinsic length or time scales, such scales must in general be negligible compared to the (translational…
The fact that not all measurements can be carried out simultaneously is a peculiar feature of quantum mechanics and responsible for many key phenomena in the theory, such as complementarity or uncertainty relations. For the special case of…
Gromov introduced two distance functions, the box distance and the observable distance, on the space of isomorphism classes of metric measure spaces and developed the convergence theory of metric measure spaces. We investigate several…
Dimensional types of metric scattered spaces are investigated. Revised proofs of Mazurkiewicz-Sierpi\'nski and Knaster-Urbanik theorems are presented. Embeddable properties of countable metric spaces are generalized onto uncountable metric…