All Questions
14 questions with no upvoted or accepted answers
10
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0
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272
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What is the smallest $\sigma$-algebra of reals that is closed under addition of sets?
What is the smallest $\sigma$-algebra $\Sigma\subseteq\mathcal P(\Bbb R)$ containing the open sets and such that if $A,B\in\Sigma$, then $$A+B=\{a+b\mid a\in A,b\in B\}\in\Sigma?$$
I know that neither ...
- 3,011
8
votes
0
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463
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When is the sigma-algebra generated by closed convex sets the same as the Borel sigma-algebra
For which topological vector spaces $E$ do we have the equality between the sigma algebra generated by the closed convex subsets of $E$ and the Borel sigma algebra of $E$ ?
More precisely, do we have ...
- 506
6
votes
0
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180
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The Fubini property of the $\sigma$-ideal generated by closed subsets of measure zero
Question. Can a Borel subset $B\subset\mathbb R\times \mathbb R$ be covered by countably many closed sets of measure zero in the plane if for every $(a,b)\in\mathbb R\times\mathbb R$ the horisontal ...
- 41.9k
5
votes
0
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214
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On generically Haar-null sets in the real line
First some definitions.
For a Polish space $X$ by $P(X)$ we denote the space of all $\sigma$-additive Borel probability measures on $X$. The space $P(X)$ carries a Polish topology generated by the ...
- 41.9k
5
votes
0
answers
138
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Disjoint covering number of an ideal
Let $\mathcal I$ be a $\sigma$-ideal with Borel base on an uncountable Polish space $X=\bigcup\mathcal I$.
Let $\mathrm{cov}(\mathcal I)$ (resp. $\mathrm{cov}_\sqcup(\mathcal I)$) be the smallest ...
- 41.9k
4
votes
0
answers
122
views
Completely I-non-measurable unions in Polish spaces
Problem. Let $X$ be a Polish space, $\mathcal I$ be a $\sigma$-ideal with Borel base, and $\mathcal A\subset\mathcal I$ be a point-finite cover of $X$. Is it true that $\mathcal A$ conatins a ...
- 4,535
4
votes
0
answers
195
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A kind of 0-1 law?
Suppose that P is a Borel subset of Baire $\times$ Baire, such that for every pair $x,x'$ of reals in the horizontal copy of Baire,
if: $x,x'$ are $E_0$-equivalent (that is, $x(n)=y(n)$ for all but ...
- 2,281
3
votes
0
answers
77
views
Can we generalize the Kuratowski Extension Theorem to Souslin spaces?
The Kuratowski Extension Theorem says: Let $(X,\mathcal{A})$ be a measurable space, $Y$ be a polish space, $A\subseteq X$, and $f:A\to Y$ be a measurable map. Then there is a measurable function $F:X\...
- 647
3
votes
0
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92
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Is there a T3½ category analogue of the density topology?
Motivation: I understand that various attempts have been made at defining a topology on $\mathbb{R}$ that is an analogue of the density topology ([1]) but for category (and meager sets) instead of ...
- 32.5k
3
votes
0
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277
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For METRIZABLE spaces, do the Banach classes and Baire classes coincide?
In this paper: 'Borel structures for Function spaces' by Robert Aumann,
http://projecteuclid.org/euclid.ijm/1255631584
Aumann claims that when X and Y are metric spaces (among other things), the ...
- 456
2
votes
0
answers
49
views
$\sigma$-compactness of probability measures under a refined topology
Denote Polish spaces $(X, \tau_x)$ and $(Y, \tau_y)$, where $X$ and $Y$ are closed subsets of $\mathbb{R}$. Consider a Borel measurable function $f: (X \times Y, \tau_x \times \tau_y) \rightarrow \...
- 195
2
votes
0
answers
102
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Is this concrete set generically Haar-null?
This question is related to this problem on MO about generically Haar-null sets in locally compact Polish groups but is more concrete.
First we recall the definition of a generically Haar-null set in ...
- 41.9k
1
vote
0
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155
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Study of the class of functions satisfying null-IVP
$\mathcal{N}_u$ : Class of all uncountable Lebesgue-null set i.e all uncountable sets having Lebesgue outer measure $0$.
Let $f:\Bbb{R}\to \Bbb{R}$ be a function with the following property :
$\...
- 307
1
vote
0
answers
155
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$f:Y\to X$ continuous with $f^{-1}(x)$ compact for $x\in X$, does there exist a Borel measurable map $g:X\to Y$?
Let $X,Y$ be Polish, metric spaces. $f:Y\to X$ is a continuous, surjective map and for any $x\in X$, $f^{-1}(x)\subset Y$ is compact. Is it true that there is a injective, Borel measurable map $g:X \...
