Introduction to Measure Space

What is/are Measure Space?

Measure Space - Our second aim is to study the existence of principal eigenvalue of the measure differential equation, and we will prove the principal eigenvalue is continuously depending on the weight measure in the weak⁎ topology of the measure space. ^[1] In this paper, we study the problem of a fair redistribution of resources among agents in an exchange economy a la Shitovitz (Econometrica 41:467–501, 1973), with agents’ measure space having both atoms and an atomless sector. ^[2] Here we present a general scheme to define localization in measure spaces, which is based on what we call Rényi occupations, from which any measure of localization can be derived. ^[3] Preference aggregation is here investigated for a society defined as a measure space of individuals and called a measure society. ^[4] A synchronous driver system wirelessly triggers the two EF sensor arrays synchronously to measure space electric fields. ^[5] We consider a semigroup acting on the function space L based a measure space. ^[6] Our setting is a discrete time dynamical system, namely the successive iterations of a measure-preserving mapping on a measure space, generalizing Hamiltonian dynamics in phase space. ^[7] Under natural assumptions about the measure spaces, the topological size as well as the algebraic size of the family of measurable real functions on the product measure space satisfying or not the conclusion of the Fubini theorem are analyzed. ^[8] We exemplify some particular cases of this general pseudometric in the contexts of measure spaces, Euclidean spaces, and fuzzy sets. ^[9] Applying medial limits we describe bounded solutions $$\varphi :S\rightarrow {\mathbb {R}}$$ φ : S → R of the functional equation $$\begin{aligned} \varphi (x)=\int _{\Omega }g(\omega )\varphi (f(x,\omega ))d\mu (\omega )+G(x), \end{aligned}$$ φ ( x ) = ∫ Ω g ( ω ) φ ( f ( x , ω ) ) d μ ( ω ) + G ( x ) , where $$(\Omega ,{\mathcal {A}},\mu )$$ ( Ω , A , μ ) is a measure space, $$S\subset \mathbb R$$ S ⊂ R , $$f:S\times \Omega \rightarrow S$$ f : S × Ω → S , $$g:\Omega \rightarrow {\mathbb {R}}$$ g : Ω → R is integrable and $$G:S\rightarrow {\mathbb {R}}$$ G : S → R is bounded. ^[10] Prior research has documented that the presence of an integer ratio is beneficial, particularly if the integer relationship is within the same measure space. ^[11] We introduce the notion of DTM-signature, a measure on R that can be associated to any metric-measure space. ^[12] I prove that the spectrum of the Laplace-Beltrami operator with the Neumann boundary condition on a compact Riemannian manifold with boundary admits a fast approximation by the spectra of suitable graph Laplacians on proximity graphs on the manifold, and similar graph approximation works for metric-measure spaces glued out of compact Riemannian manifolds of the same dimension. ^[13] Finding an approximate probability distribution best representing a sample on a measure space is one of the most basic operations in statistics. ^[14] Convex functions over measure spaces, constructed as Fenchel conjugates of integral functions of continuous functions, are shown to be sometimes equal to some integral of a function of their density. ^[15] For the algorithmic solution a class of accelerated conditional gradient methods in measure space is derived, which exploits the structural properties of the design problem to ensure convergence towards sparse solutions. ^[16] The embedded monoids are topologically distributed in the measure space. ^[17] Using a method of Korobenko, Maldonado and Rios we show a new characterization of doubling metric-measure spaces supporting Poincar\'e inequalities without assuming a priori that the measure is doubling. ^[18]

curvature dimension condition

Similarly, for metric measure spaces we study how the curvature-dimension condition in the sense of Lott–Sturm–Villani will transform under time change. ^[1] The Lott–Sturm–Villani Curvature-Dimension condition provides a synthetic notion for a metric-measure space to have Ricci-curvature bounded from below and dimension bounded from above. ^[2] We prove the stability of the Riemannian curvature dimension condition introduced by Ambrosio–Gigli–Savare under the concentration of metric measure spaces introduced by Gromov. ^[3] We study several problems concerning conformal transformation on metric measure spaces, including the Sobolev space, the differential structure and the curvature-dimension condition under conformal transformations. ^[4]

non homogeneous metric

Let ( X , d , μ ) $(\mathcal{X}, d, \mu )$ be a non-homogeneous metric measure space, which satisfies the geometrically doubling condition and the upper doubling condition. ^[1] The aim of this paper is to establish the boundedness of commutator [ b , g ˙ r ] \left[b,{\dot{g}}_{r}] generated by Littlewood-Paley g g -functions g ˙ r {\dot{g}}_{r} and b ∈ RBMO ( μ ) b\in {\rm{RBMO}}\left(\mu ) on non-homogeneous metric measure space. ^[2] Let (X,d,μ) be a non-homogeneous metric measure space satisfying the geometrically doubling condition and the upper doubling condition. ^[3]

Metric Measure Space

This paper generalizes to the context of smooth metric measure spaces and submanifolds with negative sectional curvatures some well-known geometric estimates on the p-fundamental tone by using vect. ^[1] We study the maximal operator on continuous functions in the setting of metric measure spaces. ^[2] The triangulation, viewed as a curve-decorated metric measure space equipped with its graph distance, the counting measure on vertices, and a single percolation interface converges with respect to a version of the Gromov-Hausdorff topology. ^[3] We prove a Yau’s type gradient estimate for positive f -harmonic functions with the Dirichlet boundary condition on smooth metric measure spaces with compact boundary when the infinite dimensional Bakry–Emery Ricci tensor and the weighted mean curvature are bounded below. ^[4] Let ( X , d , μ ) $(\mathcal{X}, d, \mu )$ be a non-homogeneous metric measure space, which satisfies the geometrically doubling condition and the upper doubling condition. ^[5] Inspired by recent advances in the theory of quantization for metric measure spaces, we define Quantized Gromov Wasserstein (qGW): a metric that treats parts as fundamental objects and fits into a hierarchy of theoretical upper bounds for the GW problem. ^[6] Gigli, we can extend the notion of divergence-measure vector fields $\mathcal{DM}^p(\mathbb{X})$, $1\le p \le \infty$, to the very general context of a (locally compact) metric measure space $(\mathbb{X},d,\mu)$ satisfying no further structural assumptions. ^[7] In this paper, we prove sharp gradient estimates for positive solutions to the weighted heat equation on smooth metric measure spaces with compact boundary. ^[8] It states that symmetric processes associated with resistance metric measure spaces converge when the underlying spaces converge with respect to the Gromov-Hausdorff-vague topology, and a certain uniform recurrence condition is satisfied. ^[9] Comparing with the known theory of these spaces on metric measure spaces, a major novelty of this article is that all results presented in this article get rid of the dependence on the reverse doubling assumption of the considered measure of the underlying space and hence give a final real-variable theory of these function spaces on spaces of homogeneous type. ^[10] We give a detailed proof to Gromov’s statement that precompact sets of metric measure spaces are bounded with respect to the box distance and the Lipschitz order. ^[11] In this paper we address the following parabolic equation @@ on a smooth metric measure space with Bakry–Emery curvature bounded from below for F being a differentiable function defined on $\mathbb {R}$. ^[12] Let $$(M,g,e^{-\phi } d\nu )$$ be a complete smooth metric measure space with the m-Bakry–Emery Ricci curvature bounded from below. ^[13] In this paper, we establish local and global elliptic type gradient estimates for a nonlinear parabolic equation on a smooth metric measure space whose underlying metric and potential satisfy a (k,m){(k,m)}-super Perelman–Ricci flow inequality. ^[14] In this paper, we continue our investigations giving the characterization of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. ^[15] In this paper, we first prove the $f$-mean curvature comparison in a smooth metric measure space when the Bakry-Emery Ricci tensor is bounded from below and $|f|$ is bounded. ^[16] In this paper, we study the eigenvalue problem of poly-drifting Laplacian on complete smooth metric measure space ( M , ⟨ , ⟩ , e − ϕ d v ) \left(M,\langle ,\rangle ,{e}^{-\phi }{\rm{d}}v) , with nonnegative weighted Ricci curvature Ric ϕ ≥ 0 {{\rm{Ric}}}^{\phi }\ge 0 for some ϕ ∈ C 2 ( M ) \phi \in {C}^{2}\left(M) , which is uniformly bounded from above, and successfully obtain several universal inequalities of this eigenvalue problem. ^[17] In this article, we study the Ricci flow neckpinch in the context of metric measure spaces. ^[18] The main target of this article is the boundedness of the fractional maximal operator , on Musielak–Orlicz spaces $$L^{\Phi }(X)$$ over unbounded quasi-metric measure spaces as an extension of recent results by Cruz-Uribe and Shukla (Studia Math 242(2):109–139, 2018) and the authors (2019), where $$\eta $$ is the order of the fractional maximal operator and $$\lambda $$ is its modification rate. ^[19] We assume that $$\mathfrak {M}^{n}$$ M n is an n -dimensional cigar metric measure space (CMMS for short) endowed with cigar metric and certain smooth potential on $$\mathbb {R}^{n}$$ R n. ^[20] Similarly, for metric measure spaces we study how the curvature-dimension condition in the sense of Lott–Sturm–Villani will transform under time change. ^[21] Motivated by the joint work of the first author with Khanh and Ngo, and a recent work regarding to a Liouville property of p-Lichnerowicz equation by Zhao, in this paper, we study the following elliptic equation on smooth metric measure spaces ( M , g , e − f d v ) with a weighted Poincare inequality Δ p , f v + h ( v ) = 0. ^[22] We show that, given a homeomorphism f:G→Ω{f:G\rightarrow\Omega} where G is an open subset of ℝ2{\mathbb{R}^{2}} and Ω is an open subset of a 2-Ahlfors regular metric measure space supporting a weak (1,1){(1,1)}-Poincaré inequality, it holds f∈BVloc⁡(G,Ω){f\in{\operatorname{BV_{\mathrm{loc}}}}(G,\Omega)} if and only if f-1∈BVloc⁡(Ω,G){f^{-1}\in{\operatorname{BV_{\mathrm{loc}}}}(\Omega,G)}. ^[23] Such subsets also give, in general, new examples of Sobolev extension domains on doubling metric measure spaces. ^[24] We study fractional potential of variable order on a bounded quasi-metric measure space $$(X,d,\mu )$$ as acting from variable exponent Morrey space $$ L ^{p(\cdot ), \lambda (\cdot )} (X) $$ to variable exponent Campanato space $$ \mathscr {L } ^{p(\cdot ), \lambda (\cdot )} (X) $$. ^[25] In this paper we first discuss weighted mean curvature and volume comparisons on smooth metric measure space $$(M, g, e^{-f}dv)$$ ( M , g , e - f d v ) under the integral Bakry–Émery Ricci tensor bounds. ^[26] Second: all of our results hold in the higher generality of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below in synthetic sense, the so-called RCD ( K , ∞ ) spaces. ^[27] A key ingredient is a decomposition of Hölder cut-offs in terms of an appropriate class of approximations of the identity, which we obtain on arbitrary Ahlfors-regular metric measure spaces and generalises a previous result of A. ^[28] ” This work axiomatized several aspects of Euclidean quasiconformal geometry in the setting of metric measure spaces. ^[29] In this paper we prove mean curvature comparisons and volume comparisons on a smooth metric measure space when the integral radial Bakry-Émery Ricci tensor and the potential function or its gradient are bounded. ^[30] The spaces and operators are defined on quasi-metric measure spaces with doubling condition (spaces of homogeneous type). ^[31] The aim of this paper is to establish the boundedness of commutator [ b , g ˙ r ] \left[b,{\dot{g}}_{r}] generated by Littlewood-Paley g g -functions g ˙ r {\dot{g}}_{r} and b ∈ RBMO ( μ ) b\in {\rm{RBMO}}\left(\mu ) on non-homogeneous metric measure space. ^[32] Unlike Gromov-Wasserstein (GW) distance which compares pairwise distances of elements from each distribution, we consider a method allowing to embed the metric measure spaces in a common Euclidean space and compute an optimal transport (OT) on the embedded distributions. ^[33] In the course of proving this result, we first show that on X̃, the logarithm of a reverseHölder weight w is in BMO(X̃), and that the above-mentioned connection holds on metric measure spaces X̂ := (X, d, μ). ^[34] Our method is robust enough to apply not only for $\Z^d$ but also for more general graphs whose scaling limits are nice metric measure spaces. ^[35] Spaces are defined, generally speaking, on quasi-metric measure spaces with doubling condition (spaces of homogeneous type) but the results are new even for Euclidean domains with Lebesgue measure. ^[36] We prove pointwise and $L^{p}$-gradient comparison results for solutions to elliptic Dirichlet problems defined on open subsets of a (possibly non-smooth) space with positive Ricci curvature (more precisely of an $\mathrm{RCD}(K,N)$ metric measure space, with $K>0$ and $N\in (1,\infty)$). ^[37] We establish new eigenvalue inequalities in terms of the weighted Cheeger constant for drifting p -Laplacian on smooth metric measure spaces with or without boundary. ^[38] Let (X,d,μ) be a non-homogeneous metric measure space satisfying the geometrically doubling condition and the upper doubling condition. ^[39] Let ( M n , g , e − ϕ d v ) be a smooth metric measure space. ^[40] We prove that the group of isometries of a metric measure space that satisfies the Riemannian curvature condition, $$RCD^*(K,N),$$RCD∗(K,N), is a Lie group. ^[41] We prove the stability of the Riemannian curvature dimension condition introduced by Ambrosio–Gigli–Savare under the concentration of metric measure spaces introduced by Gromov. ^[42] We study several problems concerning conformal transformation on metric measure spaces, including the Sobolev space, the differential structure and the curvature-dimension condition under conformal transformations. ^[43] We study (p-harmonic) singular functions, defined by means of upper gradients, in bounded domains in metric measure spaces. ^[44] We focus especially on a class of metric measure spaces including intersecting submanifolds of $\mathbb{R}^n$, a context in which our notion brings new insights; the Kirchhoff law appears as a special case. ^[45]

Finite Measure Space

The aim of the article is to provide a characterization of the Haagerup property for locally compact, second countable groups in terms of actions on $\unicode[STIX]{x1D70E}$ -finite measure spaces. ^[1] We establish the existence and uniqueness of strong solutions to stochastic porous media equations driven by Levy noise on a $\sigma$-finite measure space $(E,\mathcal{B}(E),\mu)$, and with the Laplacian replaced by a negative definite self-adjoint operator. ^[2] It is shown that for a dense $G_\delta $ -subset of the subgroup of non-singular transformations (of a standard infinite $\sigma $ -finite measure space) whose Poisson suspensions are non-singular, the corresponding Poisson suspensions are ergodic and of Krieger’s type III1. ^[3] In this paper we extend the construction of Grand and Small Lebesgue spaces for the case of general Banach function spaces on finite measure space. ^[4] Let $$(S, {\mathcal {B}}, m)$$ be a finite measure space. ^[5] We give the lacunary ideal form of Egorov’s theorem for sequences of fuzzy valued measurable functions defined on a finite measure space ( X , M , μ ). ^[6] In this paper, we characterize the F -transitive and the d F -transitive families of composition operators on L p ( X , B , μ ) , where ( X , B , μ ) is a σ-finite measure space. ^[7] Existence of fixed point of a Frobenius-Perron type operator P : L1 ⟶ L1 generated by a family {φy}y∈Y of nonsingular Markov maps defined on a σ-finite measure space (I, Σ, m) is studied. ^[8] Motivated by applications to stochastic programming, we introduce and study the {\em expected-integral functionals} in the form \begin{align*} \mathbb{R}^n\times \operatorname{L}^1(T,\mathbb{R}^m)\ni(x,y)\to\operatorname{E}_\varphi(x,y):=\int_T\varphi_t(x,y(t))d\mu \end{align*} defined for extended-real-valued normal integrand functions $\varphi:T\times\mathbb{R}^n\times\mathbb{R}^m\to[-\infty,\infty]$ on complete finite measure spaces $(T,\mathcal{A},\mu)$. ^[9] To avoid the difficulty of computing the normalization coefficient, such as in Exponential random graphical models (ERGMs) and Markov networks, we construct a finite measure space with measure ratio statistics. ^[10] It is also shown that the former separation result (existence of z) can be deduced from the latter one (existence of w) by using a doubly stochastic operator on the Banach space \(L^{\varrho }\left( T\right) \), where T is a finite measure space and \(\varrho \in \left[ 1,+\infty \right] \). ^[11]

Probability Measure Space

Second, the subspace optimization method in the probability measure space is integrated into the proposed SVB method to solve the involved ill-posed inverse problem. ^[1] The role of isomorphisms of Kolmogorovian probability measure spaces is specified in what we call the “Maxim of Probabilism”, which states that a necessary condition for a concept to be probabilistic is its invariance with respect to measure-theoretic isomorphisms. ^[2] org/1998/Math/MathML"> p ∈ ( 1 , ∞ ) \ { 2 } spaces on probability measure spaces. ^[3] As a generalized extension of Pawlak’s rough set model, the multigranulation decision-theoretic rough set model in ordered information systems utilizes the basic set assignment function to construct probability measure spaces through dominance relations. ^[4]

Radon Measure Space

The same holds for each quasi-regular strongly local Dirichlet space over a metrizable Luzin σ-finite Radon measure space, and admitting carré du champ operator. ^[1] We characterize the weak McShane integrability of a vector-valued function on a finite Radon measure space by means of only finite McShane partitions. ^[2]

Complete Measure Space

Let (Ω, Σ, λ) be a finite complete measure space, (E, ξ) be a sequentially complete locally convex Hausdorff space and E′ be its topological dual. ^[1] Let (A, $$\mathscr{A}$$A, µ) be a σ-finite complete measure space, and let p(·) be a µ-measurable function on A which takes values in (1, ∞). ^[2]

General Measure Space

Based on the Gale–Ryser theorem [2, 6], for the existence of suitable $(0,1)$ -matrices for different partitions of a natural number, we revisit the classical result of Lorentz [4] regarding the characterization of a plane measurable set, in terms of its cross-sections, and extend it to general measure spaces. ^[1] We derive Adams inequalities for potentials on general measure spaces, extending and improving previous results obtained by the authors. ^[2]

measure space satisfying

Under natural assumptions about the measure spaces, the topological size as well as the algebraic size of the family of measurable real functions on the product measure space satisfying or not the conclusion of the Fubini theorem are analyzed. ^[1] Let (X,d,μ) be a non-homogeneous metric measure space satisfying the geometrically doubling condition and the upper doubling condition. ^[2]