Questions tagged [borel-sets]
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113 questions
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Understanding proof that any injective Borel measurable function between Polish spaces is a Borel isomorphism onto its range (from Kechris' book)
I am trying to understand the proof of the following Corollary 15.2 in the book Classical descriptive Set Theory by Kechris.
Corollary: Let $X$, $Y$ be standard Borel spaces and $f:X\rightarrow Y$ be ...
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Understanding the theorem from Kechris' book that Borel sets are mapped to Borel sets under injective functions
Theorem 15.1 in Classical Descriptive Set Theory by Kechris states:
(i) Let $X, Y$ be Polish spaces and $f:X\rightarrow Y$ be continuous. If $A\subseteq X$ is Borel and $f|_A$ is injective, then $f(A)$...
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Can a Borel set in the plane intersect every arc but contain none?
A set $S\subseteq \mathbb{R}^2$ which intersects every arc but contains none can be constructed using transfinite recursion (just well-order the arcs then ensure each contains a point in $S$ and $\...
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Optimal complexity of Borel bijections between Polish spaces
Let $X$ and $Y$ be uncountable Polish spaces and $f:X\to Y$ a Borel bijection (which automatically has Borel inverse); such a map $f$ exists by Kuratowski's theorem.
Given countable ordinals $1\leq \...
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Locally compact, second countable, Hausdorff topology refining the right order topology
Let $A\subseteq \mathbb R$ be Borel. Is there a second countable, locally compact, Hausdorff topology on $A$ that is finer than the right order topology (i.e. the topology with base $\{x\in A:a< x\}...
- 175
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Is every Borel set a Caccioppoli set after alterations of measure zero?
Let $E\subset\mathbb R^n$ be a Borel set. Does there always exist a Caccioppoli set $F$ such that $\mathcal L^n(E\Delta F)=0$ (where $\mathcal L^n$ is the $n$-dimensional Lebesgue measure of $\mathbb ...
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Are measures singular with respect to all representing measures in $\mathbb{D}^n$ always concentrated on null-sets?
Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$.
We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if
$$\forall_{u \in A(\...
- 53
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Intersection of sigma algebras generated by shifts
EDIT: Iosif's answer showed that my motivation for this question was mislead.
To keep this question interesting for a broader readership, let us forget about sequence spaces and tail algebras and ...
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Borel complexity of the set of generic points for an invariant measure in a minimal system
I would like to know what are possible Borel complexities of the set of generic points for a minimal topological dynamical system. The only possible complexity for which we do not know if it is ...
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What properties does representing measure $\mu$ for $z\in\mathbb{D}^n$ has to satisfy so that $\nu=0$ is the only measure singular with respect to it?
Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$.
We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if
$$\forall_{u \in A(\...
- 53
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Is there an equivalent condition for Borel projections being Borel?
Let $X,Y$ be Polish spaces, and $B\subseteq X \times Y$ a Borel subset. The projection $B_X$ is not necessarily Borel in $X$. I have seen a few sufficient conditions for the projection to be Borel, ...
- 301
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Definition of "interval of continuity" for function defined on sets
At the beginning of Chapter 8 of Kubilius's Probabilistic Methods in the Theory of Numbers, the author defines $Q=Q(E)$ to be a completely additive nonnegative function defined for all Borel subsets $...
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What is an "open Baire set"?
In Measures Which Agree on Balls by Hoffmann-Jørgensen, it is stated that if $\varphi$ is a Baire function (which I presume means a pointwise limit of continuous functions), then $\{a<\varphi\}$ is ...
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Borel functions in C*-algebras
Is there a way of defining representations of separable $C^*$-algebras, say $\Phi$, so that
$\Phi(A)$ is faithful representation of $A$ on a separable Hilbert space.
There is a closure operation $A\...
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Borel sets in Vietoris topology
Let $\mathcal{K} = \mathcal{K}(\mathbb{N}^{\mathbb{N}})$ be the set of all non-empty compact subsets of the Baire space $\mathbb{N}^\mathbb{N}$ equipped with the Vietoris topology. Let $G$ be a Borel ...
