# Questions tagged [borel-sets]

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### The Borel class of a subset of $\mathbb Z^\omega$

Define $F(t)=\ln(t+1)$ for $t\geq 0$. For each sequence of integers $s=s_0s_1s_2...\in \mathbb Z^\omega$ define $$t^*_{ s}=\sup_{n\geq 0}F^{n}(|s_n|)$$ where $F^{n}$ is the $n$-fold composition of $F$...
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### separable support of Borel measure, with tau-additive measure and full support

I have a problem with Proposition 7.2.10 in Bogachev's Measure Theory Volume II book on page 77 (I have link to my drive with that book https://drive.google.com/file/d/...
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### Does there exist a non-zero signed finite borel measure which is zero on all balls?

Let $(X,d)$ be a compact separable metric space. Let $\mu$ be a Borel, regular, finite, signed measure on $X$ such that for all $x\in X$, for all $r>0$, $\mu(B(x,r))=0$, where $B$ denotes the (...
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### Borel rank collapse in Hilbert cube modulo $\sigma$-ideal generated by zero-dimensional sets

Both of the commonly studied $\sigma$-ideals (meager sets and null sets) in Polish spaces with a natural measure (i.e. $\mathbb{R}$, $[0,1]$, $[0,1]^\omega$, $2^{\omega}$, etc.) have the nice property ...
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### Disjoint union of measures

This is a sort of follow-up question to this old post I came across. Setup: Let $\{X_n\}_{n \in \mathbb{N}}$ be a collection of Hausdorff topological spaces and let $\{\Sigma_n\}_{n \in \mathbb{N}}$ ...
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### Explicit examples of (probability) measures on $\prod \mathbb{R}$

Let $\prod_{n \in \mathbb{N}} \mathbb{R}$ be equipped with the Tikhonov product of the Euclidean topologies on $\mathbb{R}$ and let $B$ the corresponding Borel $\sigma$-algebra. What is are some ...
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### Subsets of reals which are both $F_{\sigma\delta}$ and $G_{\delta\sigma}$

Let $X\subseteq \mathbb R$ such that $X$ is an $F_{\sigma\delta}$-set (in $\mathbb R$); and $X$ is a $G_{\delta\sigma}$-set. It is not necessarily true that $X$ must be $F_\sigma$ or $G_\delta$. A ...
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### Is every closed subset of finite measure contained in an open subset of finite measure?

Could someone will verify my statement: For every locally finite Borel measure on metric space and closed set $F$ with finite measure, there exists open set $U$ such that $F \subset U$ and $U$ has ...
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### Regularity of locally finite Borel measure

Do you know any proof that locally finite Borel measure on metric space is regular ? I found many proofs only for finite Borel measure, but it's not satisfies me. Or maybe do you know any books or ...
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### Abstract transverse measure theory

After reading Noncommutative Geometry book (see here) I came across the notion of the so called abstract transverse measure theory which is a generalization of standard measure theory well adapted to ...
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### Do continuous maps factor through continuous surjections via Borel maps?

Let $f \colon X \twoheadrightarrow Y$ be a continuous surjection between compact Hausdorff spaces, and $g \colon \mathbb{R} \to Y$ a continuous function. Can you always find a Borel-measurable ...
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### The space of Borel function modulo comeager sets is Dedekind complete

Let $(X,\tau)$ be a topological space. Denote by $Bor(X)$ the space of Borel functions $f:X\to\mathbb{R}$ where we identify two functions whenever they agree on the complement of a meager set. We ...
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### Measurability of a 'cone'

Let A be a (Lebesgue) measurable set in $\mathbb{R}^n$. Consider the 'cone with base A' $A(1) = \{\alpha x \in \mathbb{R}^n : x \in A, \alpha \in (0,1] \}$. Is B Lebesgue measurable? I assume it is, ...
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### Open set of geodesics implies the set of starting points is open

Let $X$ be a complete and separable metric space, let $G(X) \subset C([0,1],X)$ be the space of continuous curves from $[0,1]$ to $X$ with constant speed, i.e. $$d(f(t),f(s)) = |t-s| d(f(0), f(1)).$$...
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### Absolute continuity of probabilities on Polish spaces and open sets. [closed]

On a polish space $\mathcal{X}$ i consider two Borel probabilities $P$ and $Q$ such that for any open set $E$ of $\mathcal{X}$ we have : $P(E) =0$ implies $Q(E)=0$. Does this imply that $Q$ is ...
Does anyone knows whether the set of the absolutely continous functions $F :[0,1]\to \mathbb{R}^d$ of the form $$F(t)= a + \int_0^tf(s) ds$$ where $f$ is an integrable function is a Borel set of the ...
Let LSC = LSC([0,1]) be the set of non-negative, lower semi-continuous functions on the unit interval which take values in $\mathbb{R}_+ \cup \{\infty\}$. We use epigraph topology on LSC, i.e. a ...