Measurable function ≈ Measurable function
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Nonlocal self-improving properties Open
Solutions to nonlocal equations with measurable coefficients are higher differentiable. ¶ Specifically, we consider nonlocal integrodifferential equations with measurable coefficients whose model is given by ¶\n\n ∫ ℝn ∫ ℝn[u(x)−u(y)][η(x)…
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A convenient category for higher-order probability theory Open
Convex semilattices are algebras that are at the same time a convex algebra and a semilattice, together with a distributivity axiom. These algebras have attracted some attention in the last years as suitable algebras for probability and no…
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A convenient category for higher-order probability theory Open
Higher-order probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard measure-theoretic formalization of probabil…
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Spectral Theory for Gaussian Processes: Reproducing Kernels, Boundaries, and L<sup>2</sup>-Wavelet Generators with Fractional Scales Open
A recurrent theme in functional analysis is the interplay between the theory of positive definite functions, and their reproducing kernels, on the one hand, and Gaussian stochastic processes, on the other. This central theme is motivated b…
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Sets of uniformly absolutely continuous norm in symmetric spaces of measurable operators Open
We characterise sets of uniformly absolutely continuous norm in strongly symmetric spaces of -measurable operators. Applications are given to the study of relatively weakly compact and relatively compact sets and to compactness properties …
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Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems Open
We show that an $R^d$-topological dynamical system equipped with an invariant ergodic measure has discrete spectrum if and only it is $\mu$-mean equicontinuous (proven for $Z^d$ before). In order to do this we introduce mean equicontinuity…
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Lq-Estimates for stationary Stokes system with coefficients measurable in one direction Open
We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori [Formula: see text]-estimates for any [Formula: see text]…
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Reproducing pairs of measurable functions and partial inner product spaces Open
We continue the analysis of reproducing pairs of weakly measurable functions, which generalize continuous frames. More precisely, we examine the case where the defining measurable functions take their values in a partial inner product spac…
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Brownian motion on some spaces with varying dimension Open
In this paper, we introduce and study Brownian motion on a class of state spaces with varying dimension. Starting with a concrete case of such state spaces that models a big square with a flag pole, we construct a Brownian motion on it and…
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Bernstein–von Mises theorems for statistical inverse problems I: Schrödinger equation Open
We consider the inverse problem of determining the potential f > 0 in the partial differential equation \frac{\Delta}{2} u - fu =0 \: \mathrm{on}\: \mathcal O, \:\: u = g \: \mathrm{on} \: \partial \mathcal O, where \mathcal O is a bounded…
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Algebras of weakly symmetric functions on spaces of Lebesgue measurable functions Open
In this work, we investigate algebras of block-symmetric and weakly symmetric polynomials and analytic functions on complex Banach spaces of Lebesgue measurable functions, for which the $p$th power of the absolute value is Lebesgue integra…
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Duality and General Equilibrium Theory Under Knightian Uncertainty Open
Any dynamic or stochastic notion of a general equilibrium relies on the underlying commodity space. Under sole risk and without multiple-prior uncertainty, the usual choice is a Lebesgue space from standard measure theory. In the case of v…
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Pointwise convergence of Birkhoff averages for global observables Open
It is well-known that a strict analogue of the Birkhoff Ergodic Theorem in infinite ergodic theory is trivial; it states that for any infinite-measure-preserving ergodic system, the Birkhoff average of every integrable function is almost e…
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On the $f$-norm ergodicity of Markov processes in continuous time Open
Consider a Markov process $\\boldsymbol{\\Phi } =\\{ \\Phi (t) : t\\geq 0\\}$ evolving on a Polish space $\\mathsf{X} $. A version of the $f$-Norm Ergodic Theorem is obtained: Suppose that the process is $\\psi $-irreducible and aperiodic.…
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Algebras of symmetric and block-symmetric functions on spaces of Lebesgue measurable functions Open
In this work, we investigate algebras of symmetric and block-symmetric polynomials and analytic functions on complex Banach spaces of Lebesgue measurable functions for which the $p$th power of the absolute value is Lebesgue integrable, whe…
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$$L_q$$ L q -estimates for stationary Stokes system with coefficients measurable in one direction Open
We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a prioriẆq1-estimates for any q∈[2,∞) when the coefficients are me…
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Integrals of subharmonic functions and their differences with weight over small sets on a ray Open
Let $E$ be a measurable subset in a segment $[0,r]$ in the positive part of the real axis in the complex plane, and $U=u-v$ be the difference of subharmonic functions $u\not\equiv -\infty$ and $v\not\equiv -\infty$ on the complex plane. An…
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Sobolev boundedness and continuity for commutators of the local Hardy–Littlewood maximal function Open
Let \(\Omega\) be a subdomain in \(\mathbb{R}^n\) and \(M_\Omega\) be the local Hardy-Littlewood maximal function. In this paper, we show that both the commutator and the maximal commutator of \(M_\Omega\) are bounded and continuous from t…
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Regularity of Schrödinger's functional equation in the weak topology and moment measures Open
We study the continuity and the measurability of the solution to Schrödinger's functional equation, with respect to space, kernel and marginals, provided the space of all Borel probability measures is endowed with the weak topology. This i…
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Basic functional properties of certain scale of rearrangement‐invariant spaces Open
We define a new scale of function spaces governed by a norm‐like functional based on a combination of a rearrangement‐invariant norm, the elementary maximal function, and powers. A particular instance of such spaces surfaced recently in co…
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m-almost everywhere convergence of intuitionistic fuzzy observables induced by Borel measurable function Open
In paper [4] we studied the upper and the lower limits of sequence of intuitionistic fuzzy observables.We used an intuitionistic fuzzy state m for a definition the notion of almost everywhere convergence.We compared two concepts of m-almos…
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Pointwise ergodic theorems in symmetric spaces of measurable functions Open
For a Dunford-Schwartz operator in a fully symmetric space of measurable functions of an arbitrary measure space, we prove pointwise convergence of the conventional and weighted ergodic averages.
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Convergence of Singular Integral Operators in Weighted Lebesgue Spaces Open
In this paper, the pointwise approximation to functions f 2 L1;w ha; bi by the convo- lution type singular integral operators given in the following form: L (f; x) = Zb a f (t)K (t x) dt; x 2 ha; bi ; 2 R+0 where ha; bi stands for …
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Measurable versions of the Lov\'{a}sz Local Lemma and measurable graph colorings. Open
In this paper we investigate the extent to which the Lov\'asz Local Lemma (an important tool in probabilistic combinatorics) can be adapted for the measurable setting. In most applications, the Lov\'asz Local Lemma is used to produce a fun…
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Verifiable Conditions for Irreducibility, Aperiodicity and T-chain Property of a General Markov Chain Open
We consider in this paper Markov chains on a state space being an open subset of R n that obey the following general non linear state space model: Φt+1 = F (Φt, α(Φt, Ut+1)) , t ∈ N, where (Ut) t∈N * (each Ut ∈ R p) are i.i.d. random vecto…
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Invariant means and property $T$ of crossed products Open
Let $\\Gamma $ be a discrete group that acts on a semi-finite measure space $(\\Omega , \\mu )$ such that there is no $\\Gamma $-invariant function in $L^1(\\Omega , \\mu )$. We show that the absence of the $\\Gamma $-invariant mean on $L^…
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Equivalence of equicontinuity concepts for Markov operators derived from a Schur-like property for spaces of measures Open
Various equicontinuity properties for families of Markov operators have been – and still are – used in the study of existence and uniqueness of invariant probability for these operators, and of asymptotic stability. We prove a general resu…
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$$\mu$$-Norm of an Operator Open
Let $({\cal X},\mu)$ be a measure space. For any measurable set $Y\subset{\cal X}$ let $1_Y : {\cal X}\to{\mathbb R}$ be the indicator of $Y$ and let $\pi_Y$ be the orthogonal projector $L^2({\cal X})\ni f\mapsto\pi_Y f = 1_Y f$. For any b…
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Composition and multiplication operators between Orlicz function spaces Open
Composition operators and multiplication operators between two Orlicz function spaces are investigated. First, necessary and sufficient conditions for their continuity are presented in several forms. It is shown that, in general, the Radon…
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Weak law of large numbers for iterates of random-valued functions Open
Given a probability space $$ (\Omega , {\mathcal {A}}, P) $$ , a complete and separable metric space X with the $$ \sigma $$ -algebra $$ {\mathcal {B}} $$ of all its Borel subsets and a $$ {\mathcal {B}} \otimes {\mathcal {A}} $$ -measurab…