Fine connectedness and alpha-excessive functions

S. Ramaswamy

Displaying similar documents to “Fine connectedness and alpha-excessive functions”

Some properties of the balayage of measures on a harmonic space

Corneliu Constantinescu (1967)

Annales de l'institut Fourier

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On démontre plusieurs théorèmes concernant le balayage des mesures sur un espace harmonique satisfaisant aux axiomes de Bauer, parmi lesquels nous indiquons les suivants : a) la balayée $μ_{A \cup B}$ d’une mesure $μ$ sur la réunion $μ^{A} \lor μ^{B}$ (dans l’espace de Riesz de mesure) ; b) $ϵ_{x}^{A} \neq ϵ_{x}$ caractérise l’effilement de $A$ en $x$ ; c) il existe un potentiel fini et continue $p$ tel que pour tout ensemble $A {x | {\hat{R}}_{p} (x) < p (x)}$ est exactement l’ensemble des points où $A$ est effilé ; d) $μ^{A}$ est portée par la fermeture fine de $A$ ; e) si $A$ et $B$ sont...

Borel sets with σ-compact sections for nonseparable spaces

Petr Holický (2008)

Fundamenta Mathematicae

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We prove that every (extended) Borel subset E of X × Y, where X is complete metric and Y is Polish, can be covered by countably many extended Borel sets with compact sections if the sections $E_{x} = y \in Y : (x, y) \in E$ , x ∈ X, are σ-compact. This is a nonseparable version of a theorem of Saint Raymond. As a by-product, we get a proof of Saint Raymond’s result which does not use transfinite induction.

Borel spaces

K. P. S. Bhaskara Rao, B. V. Rao

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CONTENTSIntroduction............................................................................... 5Chapter 1. Borel spaces........................................................ 7 § 1. Borel spaces....................................................... 7 § 2. Classical descriptive set theory............................... 10 § 3. Measure and category............................................... 12 § 4. Countably generated structures.............................. 13 § 5. Product...

Consistency of the Silver dichotomy in generalised Baire space

Sy-David Friedman (2014)

Fundamenta Mathematicae

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Silver’s fundamental dichotomy in the classical theory of Borel reducibility states that any Borel (or even co-analytic) equivalence relation with uncountably many classes has a perfect set of classes. The natural generalisation of this to the generalised Baire space $κ^{κ}$ for a regular uncountable κ fails in Gödel’s L, even for κ-Borel equivalence relations. We show here that Silver’s dichotomy for κ-Borel equivalence relations in $κ^{κ}$ for uncountable regular κ is however consistent (with...

Borel classes of uniformizations of sets with large sections

Petr Holický (2010)

Fundamenta Mathematicae

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We give several refinements of known theorems on Borel uniformizations of sets with “large sections”. In particular, we show that a set B ⊂ [0,1] × [0,1] which belongs to $Σ ⁰_{α}$ , α ≥ 2, and which has all “vertical” sections of positive Lebesgue measure, has a $Π ⁰_{α}$ uniformization which is the graph of a $Σ ⁰_{α}$ -measurable mapping. We get a similar result for sets with nonmeager sections. As a corollary we derive an improvement of Srivastava’s theorem on uniformizations for Borel sets with $G_{δ}$ sections. ...

Mean values and associated measures of $δ$ -subharmonic functions

Neil A. Watson (2002)

Mathematica Bohemica

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Let $u$ be a $δ$ -subharmonic function with associated measure $μ$ , and let $v$ be a superharmonic function with associated measure $ν$ , on an open set $E$ . For any closed ball $B (x, r)$ , of centre $x$ and radius $r$ , contained in $E$ , let $ℳ (u, x, r)$ denote the mean value of $u$ over the surface of the ball. We prove that the upper and lower limits as $s, t \to 0$ with $0 < s < t$ of the quotient $(ℳ (u, x, s) - ℳ (u, x, t)) / (ℳ (v, x, s) - ℳ (v, x, t))$ , lie between the upper and lower limits as $r \to 0 +$ of the quotient $μ (B (x, r)) / ν (B (x, r))$ . This enables us to use some well-known measure-theoretic results to prove new variants...

Effective decomposition of σ-continuous Borel functions

Gabriel Debs (2014)

Fundamenta Mathematicae

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We prove that if a Δ¹₁ function f with Σ¹₁ domain X is σ-continuous then one can find a Δ¹₁ covering ${(A ₙ)}_{n \in ω}$ of X such that $f_{| A ₙ}$ is continuous for all n. This is an effective version of a recent result by Pawlikowski and Sabok, generalizing an earlier result of Solecki.

On Borel reducibility in generalized Baire space

Sy-David Friedman, Tapani Hyttinen, Vadim Kulikov (2015)

Fundamenta Mathematicae

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We study the Borel reducibility of Borel equivalence relations on the generalized Baire space $κ^{κ}$ for an uncountable κ with $κ^{< κ} = κ$ . The theory looks quite different from its classical counterpart where κ = ω, although some basic theorems do generalize.

Pointwise convergence and the Wadge hierarchy

Alessandro Andretta, Alberto Marcone (2001)

Commentationes Mathematicae Universitatis Carolinae

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We show that if $X$ is a $Σ_{1}^{1}$ separable metrizable space which is not $σ$ -compact then $C_{p}^{*} (X)$ , the space of bounded real-valued continuous functions on $X$ with the topology of pointwise convergence, is Borel- $Π_{1}^{1}$ -complete. Assuming projective determinacy we show that if $X$ is projective not $σ$ -compact and $n$ is least such that $X$ is $Σ_{n}^{1}$ then $C_{p} (X)$ , the space of real-valued continuous functions on $X$ with the topology of pointwise convergence, is Borel- $Π_{n}^{1}$ -complete. We also prove a simultaneous improvement of theorems...

The effective Borel hierarchy

M. Vanden Boom (2007)

Fundamenta Mathematicae

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Let K be a subclass of Mod() which is closed under isomorphism. Vaught showed that K is $Σ_{α}$ (respectively, $Π_{α}$ ) in the Borel hierarchy iff K is axiomatized by an infinitary $Σ_{α}$ (respectively, $Π_{α}$ ) sentence. We prove a generalization of Vaught’s theorem for the effective Borel hierarchy, i.e. the Borel sets formed by union and complementation over c.e. sets. This result says that we can axiomatize an effective $Σ_{α}$ or effective $Π_{α}$ Borel set with a computable infinitary sentence of the same complexity....

On the complexity of subspaces of $S_{ω}$

Carlos Uzcátegui (2003)

Fundamenta Mathematicae

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Let (X,τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ as a subset of the Cantor set $2^{X}$ (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel’skiĭ-Franklin space $S_{ω}$ is $F_{σ δ}$ . In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of $S_{ω}$ . We show that $S_{ω}$ has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology....