The Experts below are selected from a list of 10368 Experts worldwide ranked by ideXlab platform
Tamás Keleti - One of the best experts on this subject based on the ideXlab platform.
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Borel sets which are null or non-$\sigma$-finite for every translation invariant Measure
arXiv: Classical Analysis and ODEs, 2011Co-Authors: Márton Elekes, Tamás KeletiAbstract:We show that the set of Liouville numbers is either null or non-$\sigma$-finite with respect to every translation invariant Borel Measure on $\RR$, in particular, with respect to every Hausdorff Measure $\iH^g$ with gauge function $g$. This answers a question of D. Mauldin. We also show that some other simply defined Borel sets like non-normal or some Besicovitch-Eggleston numbers, as well as all Borel subgroups of $\RR$ that are not $F_\sigma$ possess the above property. We prove that, apart from some trivial cases, the Borel class, Hausdorff or packing dimension of a Borel set with no such Measure on it can be arbitrary.
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is lebesgue Measure the only sigma finite invariant Borel Measure
arXiv: Classical Analysis and ODEs, 2011Co-Authors: Márton Elekes, Tamás KeletiAbstract:R.D.Mauldin asked if every translation invariant $\sigma$-finite Borel Measure on $\RR^d$ is a constant multiple of Lebesgue Measure. The aim of this paper is to show that the answer is "yes and no", since surprisingly the answer depends on what we mean by Borel Measure and by constant. We present Mauldin's proof of what he called a folklore result, stating that if the Measure is only defined for Borel sets then the answer is affirmative. Then we show that if the Measure is defined on a $\sigma$-algebra \emph{containing} the Borel sets then the answer is negative. However, if we allow the multiplicative constant to be infinity, then the answer is affirmative in this case as well. Moreover, our construction also shows that an isometry invariant $\sigma$-finite Borel Measure (in the wider sense) on $\RR^d$ can be non-$\sigma$-finite when we restrict it to the Borel sets.
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Borel sets which are null or non- σ -finite for every translation invariant Measure
Advances in Mathematics, 2006Co-Authors: Márton Elekes, Tamás KeletiAbstract:Abstract We show that the set of Liouville numbers is either null or non- σ -finite with respect to every translation invariant Borel Measure on R , in particular, with respect to every Hausdorff Measure H g with gauge function g . This answers a question of R.D. Mauldin. We also show that some other simply defined Borel sets like non-normal or some Besicovitch–Eggleston numbers, as well as all Borel subgroups of R that are not F σ possess the above property. We prove that, apart from some trivial cases, the Borel class, Hausdorff or packing dimension of a Borel set with no such Measure on it can be arbitrary.
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Is Lebesgue Measure the only σ-finite invariant Borel Measure?
Journal of Mathematical Analysis and Applications, 2006Co-Authors: Márton Elekes, Tamás KeletiAbstract:AbstractS. Saks and recently R.D. Mauldin asked if every translation invariant σ-finite Borel Measure on Rd is a constant multiple of Lebesgue Measure. The aim of this paper is to investigate the versions of this question, since surprisingly the answer is “yes and no,” depending on what we mean by Borel Measure and by constant. According to a folklore result, if the Measure is only defined for Borel sets, then the answer is affirmative. We show that if the Measure is defined on a σ-algebra containing the Borel sets, then the answer is negative. However, if we allow the multiplicative constant to be infinity, then the answer is affirmative in this case as well. Moreover, our construction also shows that an isometry invariant σ-finite Borel Measure (in the wider sense) on Rd can be non-σ-finite when we restrict it to the Borel sets
Márton Elekes - One of the best experts on this subject based on the ideXlab platform.
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Borel sets which are null or non-$\sigma$-finite for every translation invariant Measure
arXiv: Classical Analysis and ODEs, 2011Co-Authors: Márton Elekes, Tamás KeletiAbstract:We show that the set of Liouville numbers is either null or non-$\sigma$-finite with respect to every translation invariant Borel Measure on $\RR$, in particular, with respect to every Hausdorff Measure $\iH^g$ with gauge function $g$. This answers a question of D. Mauldin. We also show that some other simply defined Borel sets like non-normal or some Besicovitch-Eggleston numbers, as well as all Borel subgroups of $\RR$ that are not $F_\sigma$ possess the above property. We prove that, apart from some trivial cases, the Borel class, Hausdorff or packing dimension of a Borel set with no such Measure on it can be arbitrary.
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is lebesgue Measure the only sigma finite invariant Borel Measure
arXiv: Classical Analysis and ODEs, 2011Co-Authors: Márton Elekes, Tamás KeletiAbstract:R.D.Mauldin asked if every translation invariant $\sigma$-finite Borel Measure on $\RR^d$ is a constant multiple of Lebesgue Measure. The aim of this paper is to show that the answer is "yes and no", since surprisingly the answer depends on what we mean by Borel Measure and by constant. We present Mauldin's proof of what he called a folklore result, stating that if the Measure is only defined for Borel sets then the answer is affirmative. Then we show that if the Measure is defined on a $\sigma$-algebra \emph{containing} the Borel sets then the answer is negative. However, if we allow the multiplicative constant to be infinity, then the answer is affirmative in this case as well. Moreover, our construction also shows that an isometry invariant $\sigma$-finite Borel Measure (in the wider sense) on $\RR^d$ can be non-$\sigma$-finite when we restrict it to the Borel sets.
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Borel sets which are null or non- σ -finite for every translation invariant Measure
Advances in Mathematics, 2006Co-Authors: Márton Elekes, Tamás KeletiAbstract:Abstract We show that the set of Liouville numbers is either null or non- σ -finite with respect to every translation invariant Borel Measure on R , in particular, with respect to every Hausdorff Measure H g with gauge function g . This answers a question of R.D. Mauldin. We also show that some other simply defined Borel sets like non-normal or some Besicovitch–Eggleston numbers, as well as all Borel subgroups of R that are not F σ possess the above property. We prove that, apart from some trivial cases, the Borel class, Hausdorff or packing dimension of a Borel set with no such Measure on it can be arbitrary.
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Is Lebesgue Measure the only σ-finite invariant Borel Measure?
Journal of Mathematical Analysis and Applications, 2006Co-Authors: Márton Elekes, Tamás KeletiAbstract:AbstractS. Saks and recently R.D. Mauldin asked if every translation invariant σ-finite Borel Measure on Rd is a constant multiple of Lebesgue Measure. The aim of this paper is to investigate the versions of this question, since surprisingly the answer is “yes and no,” depending on what we mean by Borel Measure and by constant. According to a folklore result, if the Measure is only defined for Borel sets, then the answer is affirmative. We show that if the Measure is defined on a σ-algebra containing the Borel sets, then the answer is negative. However, if we allow the multiplicative constant to be infinity, then the answer is affirmative in this case as well. Moreover, our construction also shows that an isometry invariant σ-finite Borel Measure (in the wider sense) on Rd can be non-σ-finite when we restrict it to the Borel sets
Valentino Magnani - One of the best experts on this subject based on the ideXlab platform.
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on a Measure theoretic area formula
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2015Co-Authors: Valentino MagnaniAbstract:We review some classical differentiation theorems for Measures, showing how they can be turned into an integral representation of a Borel Measure with respect to a fixed Caratheodory Measure. We focus our attention on the case when this Measure is the spherical Hausdorff Measure, giving a metric Measure area formula. Our aim is to use certain covering derivatives as ‘generalized densities’. Some consequences for the sub-Riemannian Heisenberg group are also pointed out.
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on a Measure theoretic area formula
arXiv: Metric Geometry, 2014Co-Authors: Valentino MagnaniAbstract:We review some classical differentiation theorems for Measures, showing how they can be turned into an integral representation of a Borel Measure with respect to a fixed Carath\'eodory Measure. We focus our attention on the case this Measure is the spherical Hausdorff Measure, giving a metric Measure area formula. Our point consists in using certain covering derivatives as "generalized densities". Some consequences for the sub-Riemannian Heisenberg group are also pointed out.
Augusto C Ponce - One of the best experts on this subject based on the ideXlab platform.
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A representation formula for the distributional normal derivative
Revista Matemática Complutense, 2020Co-Authors: Augusto C Ponce, Nicolas WilmetAbstract:We prove an integral representation formula for the distributional normal derivative of solutions of $$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} - \Delta u + V u &{}= \mu &{}&{} \text {in }\Omega , \\ u &{}= 0 &{}&{} \text {on }\partial \Omega , \end{aligned} \end{array}\right. } \end{aligned}$$ - Δ u + V u = μ in Ω , u = 0 on ∂ Ω , where $$V \in L_\mathrm {loc}^1(\Omega )$$ V ∈ L loc 1 ( Ω ) is a nonnegative function and $$\mu $$ μ is a finite Borel Measure on $$\Omega $$ Ω . As an application, we show that the Hopf lemma holds almost everywhere on $$\partial \Omega $$ ∂ Ω when $$V$$ V is a nonnegative Hopf potential.
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a representation formula for the distributional normal derivative
arXiv: Analysis of PDEs, 2020Co-Authors: Augusto C Ponce, Nicolas WilmetAbstract:We prove an integral representation formula for the distributional normal derivative of solutions of $$ \left\{ \begin{aligned} - \Delta u + V u &= \mu && \text{in $\Omega$,} u &= 0 && \text{on $\partial\Omega$,} \end{aligned} \right. $$ where $V \in L_{\mathrm{loc}}^1(\Omega)$ is a nonnegative function and $\mu$ is a finite Borel Measure on $\Omega$. As an application, we show that the Hopf lemma holds almost everywhere on $\partial\Omega$ when $V$ is a nonnegative Hopf potential.
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on the nonexistence of green s function and failure of the strong maximum principle
Journal de Mathématiques Pures et Appliquées, 2020Co-Authors: Luigi Orsina, Augusto C PonceAbstract:Abstract Given any Borel function V : Ω → [ 0 , + ∞ ] on a smooth bounded domain Ω ⊂ R N , we establish that the strong maximum principle for the Schrodinger operator − Δ + V in Ω holds in each Sobolev-connected component of Ω ∖ Z , where Z ⊂ Ω is the set of points which cannot carry a Green's function for − Δ + V . More generally, we show that the equation − Δ u + V u = μ in Ω involving a nonnegative finite Borel Measure μ has a distributional solution in W 0 1 , 1 ( Ω ) if and only if μ ( Z ) = 0 .
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on the nonexistence of green s function and failure of the strong maximum principle
arXiv: Analysis of PDEs, 2018Co-Authors: Luigi Orsina, Augusto C PonceAbstract:Given any Borel function $V : \Omega \to [0, +\infty]$ on a smooth bounded domain $\Omega \subset \mathbb{R}^{N}$, we establish that the strong maximum principle for the Schr\"odinger operator $-\Delta + V$ in $\Omega$ holds in each Sobolev-connected component of $\Omega \setminus Z$, where $Z \subset \Omega$ is the set of points which cannot carry a Green's function for $- \Delta + V$. More generally, we show that the equation $- \Delta u + V u = \mu$ has a distributional solution in $W_{0}^{1, 1}(\Omega)$ for a nonnegative finite Borel Measure $\mu$ if and only if $\mu(Z) = 0$.
Jean Ludwig - One of the best experts on this subject based on the ideXlab platform.
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fourier stieltjes algebra of a topological group
Advances in Mathematics, 2012Co-Authors: Anthony Toming Lau, Jean LudwigAbstract:Abstract This paper is an invitation to the study of the Fourier–Stieltjes algebra B ( G ) , the linear span of the continuous positive definite complex-valued functions on a topological group G . We study B ( G ) , when G has a host algebra or a group C ⁎ -algebra, the analogue of the group C ⁎ -algebra of a locally compact group. Our main challenge is that a topological group G cannot have a positive regular Borel Measure which is left translation invariant unless G is locally compact.