On the weak $L^1$ space and singular measures

Robert Kaufman

Displaying similar documents to “On the weak $L^{1}$ space and singular measures”

Singular measures and the key of G.

Stephen M. Buckley, Paul MacManus (2000)

Publicacions Matemàtiques

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We construct a sequence of doubling measures, whose doubling constants tend to 1, all for which kill a G set of full Lebesgue measure.

On certain singular measures

R. Kaufman (1968)

Colloquium Mathematicae

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Constructions of singular measures with remarkable convolution properties

Louis Pigno, Sadahiro Saeki (1981)

Studia Mathematica

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Symmetric and Zygmund measures in several variables

Evgueni Doubtsov, Artur Nicolau (2002)

Annales de l’institut Fourier

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Let $ω : (0, \infty) \to (0, \infty)$ be a gauge function satisfying certain mid regularity conditions. A (signed) finite Borel measure $μ \in ℝ^{n}$ is called $ω$ -Zygmund if there exists a positive constant $C$ such that $| μ (Q_{+}) - μ (Q_{-}) | \leq C ω (ℓ (Q_{+})) | Q_{+} |$ for any pair $Q_{+}, Q_{-} \subset ℝ^{n}$ of adjacent cubes of the same size. Similarly, $μ$ is called an $ω$ - symmetric measure if there exists a positive constant $C$ such that $| μ (Q_{+}) / μ (Q_{-}) - 1 | \leq C ω (ℓ (Q_{+}))$ for any pair $Q_{+}, Q_{-} \subset ℝ^{n}$ of adjacent cubes of the same size, $ℓ (Q_{+}) = ℓ (Q_{-}) < 1$ . We characterize Zygmund and symmetric measures in terms of their harmonic extensions. Also, we show that the quadratic...

Weak-star continuous homomorphisms and a decomposition of orthogonal measures

B. J. Cole, Theodore W. Gamelin (1985)

Annales de l'institut Fourier

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We consider the set $S (μ)$ of complex-valued homomorphisms of a uniform algebra $A$ which are weak-star continuous with respect to a fixed measure $μ$ . The $μ$ -parts of $S (μ)$ are defined, and a decomposition theorem for measures in $A^{⊥} \cap L^{1} (μ)$ is obtained, in which constituent summands are mutually absolutely continuous with respect to representing measures. The set $S (μ)$ is studied for $T$ -invariant algebras on compact subsets of the complex plane and also for the infinite polydisc algebra.

Conical measures and vector measures

Igor Kluvánek (1977)

Annales de l'institut Fourier

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Every conical measure on a weak complete space $E$ is represented as integration with respect to a $σ$ -additive measure on the cylindrical $σ$ -algebra in $E$ . The connection between conical measures on $E$ and $E$ -valued measures gives then some sufficient conditions for the representing measure to be finite.

Convolution of singular measures

Raouf Doss (1973)

Studia Mathematica

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On the complexity of sums of Dirichlet measures

Sylvain Kahane (1993)

Annales de l'institut Fourier

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Let $M$ be the set of all Dirichlet measures on the unit circle. We prove that $M + M$ is a non Borel analytic set for the weak* topology and that $M + M$ is not norm-closed. More precisely, we prove that there is no weak* Borel set which separates $M + M$ from $D^{⊥}$ (or even $L_{0}^{⊥})$ , the set of all measures singular with respect to every measure in $M$ . This extends results of Kaufman, Kechris and Lyons about $D^{⊥}$ and $H^{⊥}$ and gives many examples of non Borel analytic sets.

Displaying similar documents to “On the weak L 1 space and singular measures”

Displaying similar documents to “On the weak $L^{1}$ space and singular measures”