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Given a compact space K, we denote by P(K) the space of all Radon probability measures on K, equipped with the weak* topology inherited from C(K)*. For nonmetrizable compacta K even basic properties of P(K) spaces depend of additional axioms of set theory. We discuss here older and quite recent results on the subject. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of duality in the linear Kantorovich problem with a fixed barycenter. It is proved that Kantorovich duality holds for general lower semicontinuous cost functions on completely regular spaces. In the course of considering this subject, the question of representation of a continuous linear functional by a Radon measure is raised and solved, provided that the barycenter of the functional is given by a Radon measure. In addition, we consider two new barycentric optimization problems and prove duality results for them. [ABSTRACT FROM AUTHOR]

In the second part of his fifth problem Hilbert asks for functional equations "In how far are the assertions which we can make in the case of differentiable functions true under proper modifications without this assumption." In the case of the general functional equation f (x) = h (x , y , (g 1 (x , y)) , ... , (g n (x , y))) for the unknown function f under natural condition for the given functions it is proved on compact manifolds that f ∈ C - 1 implies f ∈ C ∞ and practically the general case can also be treated. The natural conditions imply that the dimension of x cannot be larger than the dimension of y. If we remove this condition, then we have to add another condition. In this survey paper a new problem for this second case is formulated and results are summarised for both cases. [ABSTRACT FROM AUTHOR]

We prove that the set of finite Borel measures on a separable and directionally limited metric space (X , d) is complete with respect to the metric d A (μ , ν) = sup A ∈ A μ (A) - ν (A) for all families of Borel sets A that contain every closed ball of X. This allows to prove the existence and uniqueness of the invariant Borel probability measure of certain Markov processes on X. A natural application is a Markov process induced by a random similitude. [ABSTRACT FROM AUTHOR]

This paper presents a constructive proof of the existence of a regular non-atomic strictly-positive measure on any second-countable non-atomic locally compact Hausdorff space. This construction involves a sequence of finitely-additive set functions defined recursively on an ascending sequence of rings of subsets with a set function limit that is extendable to a measure with the desired properties. Non-atomicity of the space provides a meticulous way to ensure that the set function limit is σ-additive.

A thin set is defined to be an uncountable dense zero-dimensional subset of measure zero and Hausdorff measure zero of an Euclidean space. A fat set is defined to be an uncountable dense path-connected subset of an Euclidean space which has full measure, that is, its complement has measure zero. While there are well-known pathological maps of a set of measure zero, such as the Cantor set, onto an interval, we show that the standard addition on $\mathbb {R}$ maps a thin set onto a fat set; in fact the fat set is all of $\mathbb {R}$. Our argument depends on the theorem of Paul Erdős that every real number is a sum of two Liouville numbers. Our thin set is the set $\mathcal {L}^{2}$ , where $\mathcal {L}$ is the set of all Liouville numbers, and the fat set is $\mathbb {R}$ itself. Finally, it is shown that $\mathcal {L}$ and $\mathcal {L}^{2}$ are both homeomorphic to $\mathbb {P}$ , the space of all irrational numbers. [ABSTRACT FROM AUTHOR]

In this paper, a definition of a special type of function based on the concept of proximity relation is presented in conjunction with central set theory to find a special definition of continuity between center topological spaces and the transfer of topological properties, which we call sharing central continuous function. [ABSTRACT FROM AUTHOR]

Topological measures and deficient topological measures are defined on open and closed subsets of a topological space, generalize regular Borel measures, and correspond to (nonlinear in general) functionals that are linear on singly generated subalgebras or singly generated cones of functions. They lack subadditivity, and many standard techniques of measure theory and functional analysis do not apply to them. Nevertheless, we show that many classical results of probability theory hold for topological and deficient topological measures. In particular, we prove a version of Aleksandrov's theorem for equivalent definitions of weak convergence of deficient topological measures. We also prove a version of Prokhorov's theorem which relates the existence of a weakly convergent subsequence in any sequence in a family of topological measures to the characteristics of being a uniformly bounded in variation and uniformly tight family. We define Prokhorov and Kantorovich–Rubenstein metrics and show that convergence in either of them implies weak convergence of (deficient) topological measures on metric spaces. We also generalize many known results about various dense and nowhere dense subsets of deficient topological measures. The present paper constitutes a necessary step to further research in probability theory and its applications in the context of (deficient) topological measures and corresponding nonlinear functionals. [ABSTRACT FROM AUTHOR]

We deal with the following general version of the classical moment problem: when can a linear functional on a unital commutative real algebra A be represented as an integral with respect to a Radon measure on the character space X(A) of A equipped with the Borel σ -algebra generated by the weak topology? We approach this problem by constructing X(A) as a projective limit of the character spaces of all finitely generated unital subalgebras of A. Using some fundamental results for measures on projective limits of measurable spaces, we determine a criterion for the existence of an integral representation of a linear functional on A with respect to a measure on the cylinder σ -algebra on X(A) (resp. a Radon measure on the Borel σ -algebra on X(A)) provided that for any finitely generated unital subalgebra of A the corresponding moment problem is solvable. We also investigate how to localize the support of representing measures for linear functionals on A. These results allow us to establish infinite dimensional analogues of the classical Riesz-Haviland and Nussbaum theorems as well as a representation theorem for linear functionals non-negative on a "partially Archimedean" quadratic module of A. Our results in particular apply to the case when A is the algebra of polynomials in infinitely many variables or the symmetric tensor algebra of a real infinite dimensional vector space, providing a unified setting which enables comparisons between some recent results for these instances of the moment problem. [ABSTRACT FROM AUTHOR]

We establish a connection between the function space BMO and the theory of quasisymmetric mappings on spaces of homogeneous type X ~ : = (X , ρ , μ) . The connection is that the logarithm of the generalised Jacobian of an η -quasisymmetric mapping f : X ~ → X ~ is always in BMO (X ~) . In the course of proving this result, we first show that on X ~ , the logarithm of a reverse-Hölder weight w is in BMO (X ~) , and that the above-mentioned connection holds on metric measure spaces X ^ : = (X , d , μ) . Furthermore, we construct a large class of spaces (X , ρ , μ) to which our results apply. Among the key ingredients of the proofs are suitable generalisations to (X , ρ , μ) from the Euclidean or metric measure space settings of the Calderón–Zygmund decomposition, the Vitali Covering Theorem, the Radon–Nikodym Theorem, a lemma which controls the distortion of sets under an η -quasisymmetric mapping, and a result of Heinonen and Koskela which shows that the volume derivative of an η -quasisymmetric mapping is a reverse-Hölder weight. [ABSTRACT FROM AUTHOR]

A new kind of open maps namely [αrps—·open, and Robustly αrps —· open] were presented in this paper, and the relationships, between these open maps are also studied and discussed. [ABSTRACT FROM AUTHOR]

We introduce the notion of variational measure with respect to a derivation basis in a topological measure space and consider a Kurzweil–Henstock-type integral related to this basis. We prove a version of Hake's theorem in terms of a variational measure. [ABSTRACT FROM AUTHOR]

In analogy with the lower Assouad dimensions of a set, we study the lower Assouad dimensions of a measure. As with the upper Assouad dimensions, the lower Assouad dimensions of a measure provide information about the extreme local behaviour of the measure. We study the connection with other dimensions and with regularity properties. In particular, the quasi-lower Assouad dimension is dominated by the infimum of the measure's lower local dimensions. Although strict inequality is possible in general, equality holds for the class of self-similar measures of finite type. This class includes all self-similar, equicontractive measures satisfying the open set condition, as well as certain "overlapping" self-similar measures, such as Bernoulli convolutions with contraction factors that are inverses of Pisot numbers. We give lower bounds for the lower Assouad dimension for measures arising from a Moran construction, prove that self-affine measures are uniformly perfect and have positive lower Assouad dimension, prove that the Assouad spectrum of a measure converges to its quasi-Assouad dimension and show that coincidence of the upper and lower Assouad dimension of a measure does not imply that the measure is s-regular. [ABSTRACT FROM AUTHOR]

In this paper, we investigate to what extent the conclusion of the Lebesgue dominated convergence theorem holds if the assumption of dominance is dropped. Specifically, we study both topological and algebraic genericity of the family of all null sequences of functions that, being continuous on a locally compact space and integrable with respect to a given Borel measure in it, are not controlled by an integrable function. [ABSTRACT FROM AUTHOR]

Topological measures and deficient topological measures generalize Borel measures and correspond to certain non-linear functionals. We study integration with respect to deficient topological measures on locally compact spaces. Such an integration over sets yields a new deficient topological measure if we integrate a nonnegative continuous vanishing at infinity function; and it produces a signed deficient topological measure if we integrate a continuous function on a compact space. We present many properties of these resulting deficient topological measures and of signed deficient topological measures. In particular, they are absolutely continuous with respect to the original deficient topological measure, and their corresponding non-linear functionals are Lipschitz continuous. Deficient topological measures obtained by integration over sets can also be obtained from non-linear functionals. We show that for a deficient topological measure μ that assumes finitely many values, there is a function f such that ∫ X $\begin{array}{} \int\limits_X \end{array}$ f dμ = 0, but ∫ X $\begin{array}{} \int\limits_X \end{array}$ (–f) dμ ≠ 0. We present different criteria for ∫ X $\begin{array}{} \int\limits_X \end{array}$ f dμ = 0. We also prove some convergence results, including a Monotone convergence theorem. [ABSTRACT FROM AUTHOR]

We study Riemann–Lebesgue integrability of a vector function relative to an arbitrary non-negative set function. We obtain some classical integral properties. Results regarding the continuity properties of the integral and relationships among Riemann–Lebesgue, Birkhoff simple and Gould integrabilities are also established. [ABSTRACT FROM AUTHOR]

For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain multifractal measures. However, we primarily show that they associate a new zeta function, the topological zeta function, to a fractal string in order to take into account the topology of its fractal boundary. This expands upon the geometric information garnered by the traditional geometric zeta function of a fractal string in the theory of complex dimensions. In particular, one can distinguish between a fractal string whose boundary is the classical Cantor set, and one whose boundary has a single limit point but has the same sequence of lengths as the complement of the Cantor set. Later work will address related, but somewhat different, approaches to multifractals themselves, via zeta functions, partly motivated by the present paper.

The class of stochastically self-similar sets contains many famous examples of random sets, for example, Mandelbrot percolation and general fractal percolation. Under the assumption of the uniform open set condition and some mild assumptions on the iterated function systems used, we show that the quasi-Assouad dimension of self-similar random recursive sets is almost surely equal to the almost sure Hausdorff dimension of the set. We further comment on random homogeneous and V -variable sets and the removal of overlap conditions. [ABSTRACT FROM AUTHOR]

The Besicovitch covering property (BCP) is known to be one of the fundamental tools in measure theory, and more generally, a useful property for numerous purposes in analysis and geometry. We prove both sufficient and necessary criteria for the validity of BCP in the first Heisenberg group equipped with a homogeneous distance. Beyond recovering all previously known results about the validity or non-validity of BCP in this setting, we get simple descriptions of new large classes of homogeneous distances satisfying BCP. We also obtain a full characterization of rotationally invariant distances for which BCP holds in the first Heisenberg group under mild regularity assumptions about their unit sphere. [ABSTRACT FROM AUTHOR]

We introduce canonical measures on a locally finite simplicial complex K and study their asymptotic behavior under infinitely many barycentric subdivisions. We prove that the simplices of each dimension in Sd d (K) equidistribute in |K| with respect to the Lebesgue measure as d grows to + ∞ and then prove that their asymptotic link and dual block are universal. [ABSTRACT FROM AUTHOR]

We study the non-asymptotic behavior of Coulomb gases in dimension two and more. Such gases are modeled by an exchangeable Boltzmann–Gibbs measure with a singular two-body interaction. We obtain concentration of measure inequalities for the empirical distribution of such gases around their equilibrium measure, with respect to bounded Lipschitz and Wasserstein distances. This implies macroscopic as well as mesoscopic convergence in such distances. In particular, we improve the concentration inequalities known for the empirical spectral distribution of Ginibre random matrices. Our approach is remarkably simple and bypasses the use of renormalized energy. It crucially relies on new inequalities between probability metrics, including Coulomb transport inequalities which can be of independent interest. Our work is inspired by the one of Maïda and Maurel-Segala, itself inspired by large deviations techniques. Our approach allows to recover, extend, and simplify previous results by Rougerie and Serfaty. [ABSTRACT FROM AUTHOR]

The space of real Borel measures M(S) on a metric space S under the flat norm (dual bounded Lipschitz norm), ordered by the cone M+(S) of nonnegative measures, is considered from an ordered normed vector space perspective in order to apply the well-developed theory of this area. The flat norm is considered in place of the variation norm because subsets of M+(S) are compact and semiflows on M+(S) are continuous under much weaker conditions. In turn, the flat norm offers new challenges because M(S) is rarely complete and M+(S) is only complete if S is complete. As illustrations serve the eigenvalue problem for bounded additive and order-preserving homogeneous maps on M+(S) and continuous semiflows. Both topics prepare for a dynamical systems theory on M+(S). [ABSTRACT FROM AUTHOR]

In this paper we consider the nonautonomous semilinear viscoelastic equation u t t − Δ u + ∫ 0 τ k ( s ) Δ u ( t − s ) d s + f ( x , u ) = g , τ ∈ { t , ∞ } , in R + × Ω , with Dirichlet boundary conditions and finite ( τ = t ) or infinite ( τ = ∞ ) memory. Here Ω is a bounded domain in R n with smooth boundary and the nonlinearity f : Ω × R + → R is analytic in the second variable, uniformly with respect to the first one. For this equation, we derive an appropriate Lyapunov function and we use the Łojasiewicz–Simon inequality to show that the dissipation given by the memory term is strong enough to prove the convergence to a steady state for any global bounded solution. In addition, we discuss the rate of convergence to equilibrium which is polynomial or exponential, depending on the Łojasiewicz exponent and the decay of the time-dependent right-hand side g . [ABSTRACT FROM AUTHOR]

In this paper we present some results concerning Gould integrability of vector functions with respect to a monotone measure on finitely purely atomic measure spaces. As an application a Radon-Nikodym theorem in this setting is obtained. [ABSTRACT FROM AUTHOR]

One considers Hilbert space valued measures on the Borel sets of a compact metric space. A natural numerical valued integral of vector valued continuous functions with respect to vector valued measures is defined. Using this integral, different norms (we called them Monge-Kantorovich norm, modified Monge-Kantorovich norm and Hanin norm) on the space of measures are introduced, generalizing the theory of (weak) convergence for probability measures on metric spaces. These norms introduce new (equivalent) metrics on the initial compact metric space. [ABSTRACT FROM AUTHOR]

DT-measures generalize the topological measures of Aarnes. We continue to study an integral of a real-valued continuous function on compact Hausdorff space with respect to a DT-measure and investigate the following: multiplicativity of this integral, $$w*$$ -topology on the family of DT-measures and density of various subfamilies of this space, and also integral as a set function. [ABSTRACT FROM AUTHOR]

For all continuous function g having a specific form that we call with increasing visibility, we construct a function f whose multifractal spectrum is such that $$ d_f =g\circ f$$ . The function f is obtained as an infinite superposition of piecewise $$C^1$$ functions, is also with increasing visibility, and is homogeneously multifractal; i.e., its restriction on any subinterval of $$[0,1]$$ has the same multifractal spectrum as the function f itself. In particular, we prove the existence of a function f which is its own multifractal spectrum; i.e., $$f=d_f$$ . [ABSTRACT FROM AUTHOR]

We study the long-time behavior as time goes to infinity of global bounded solutions to the following nonautonomous semilinear viscoelastic equation: in $${\mathbb {R}}^+\times \Omega $$ , with Dirichlet boundary conditions, where $$\Omega $$ is a bounded domain in $${\mathbb {R}}^n$$ and the nonlinearity f is analytic. Based on an appropriate (perturbed) new Lyapunov function and the Łojasiewicz-Simon inequality we prove that any global bounded solution converges to a steady state. We discuss also the rate of convergence which is polynomial or exponential, depending on the Łojasiewicz exponent and the decay of the term g. [ABSTRACT FROM AUTHOR]

Wave propagation in heterogeneous solids has been an interest of researchers due to industrial applications. Some of the heterogeneous materials can exhibit power law scaling in material behavior which can be characterized by the fractal dimension of the microstructure. In this study, wave propagation in heterogeneous media with self-similar structure is investigated via fractional calculus along with space-time discontinuous Galerkin method. One and two dimensional problems are studied to demonstrate the capability of the proposed model in modeling heterogeneous media. The results show that the proposed model is a good candidate for modeling the mechanical behavior of disordered materials. [ABSTRACT FROM AUTHOR]

Assuming b = c (or some weaker statement), we construct a compactification γω of ω such that its remainder γ ω ∖ ω is nonseparable and carries a strictly positive measure. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and uniqueness of a solution for a second-order set-valued differential equation with three-point boundary conditions, where the perturbation is measurable with respect to the time variable and Lipschitzian with respect to the second and third variables. [ABSTRACT FROM AUTHOR]

We give an explicit description of all finite Borel measures on Noetherian topological spaces X, and characterize them as objects dual to a space of functions on X. We use these results to study the asymptotic behavior of continuous dynamical systems on Noetherian spaces. [ABSTRACT FROM AUTHOR]