All Questions
11
questions
10
votes
0
answers
248
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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 ...
- 2,672
5
votes
1
answer
240
views
How complex is the orbit equivalence relation of $\mathrm{Iso}_0(X)\curvearrowright S_X$ for $X=L^p([0,1])$?
For a Banach space $X$ let $S_X$ denote its unit sphere and let $\mathrm{Iso}_0(X)$ denote the group of rotations of $X$, that is isometries fixing the origin. There is a natural continuous action $\...
- 9,145
4
votes
2
answers
580
views
Question about additive subgroups of the real line and the density topology
I am new studying additive subgroups of the real line, I would like to know if someone could give me an idea for the next question.
Let $m$ be the Lebesgue measure in $\mathbb{R}$. A measurable set $E\...
- 551
1
vote
1
answer
248
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CH and the density topology on $\mathbb{R}$
In the article AN EXAMPLE INVOLVING BAIRE SPACES (https://www.ams.org/journals/proc/1975-048-01/S0002-9939-1975-0362249-1/S0002-9939-1975-0362249-1.pdf) of H. E. White Jr. it is shown that, assuming ...
- 9,540
12
votes
0
answers
171
views
A connected Borel subgroup of the plane
It is known that the complex plane $\mathbb C$ contain dense connected (additive) subgroups with dense complement but each dense path-connected subgroup of $\mathbb C$ necessarily coincides with $\...
- 40.2k
2
votes
0
answers
102
views
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 ...
- 40.2k
5
votes
0
answers
212
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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 ...
- 40.2k
3
votes
0
answers
142
views
Is an Abelian topological group compact if it is complete and Bohr-compact?
A topological group $G$ will be called Bohr-compact if its Bohr topology (i.e., the largest precompact group topology) is compact and Hausdorff.
A topological group $G$ is Bohr-compact if it admits ...
- 280
4
votes
1
answer
315
views
Is there a topologizable group admitting only Raikov-complete group topologies?
Definition. A group $G$ is called complete (resp. non-topologizable) if each Hausdorff group topology on $G$ is Raikov-complete (resp. discrete). It is clear that each non-topologizable group is ...
- 40.2k
12
votes
0
answers
365
views
Does each compact topological group admit a discontinuous homomorphism to a Polish group?
A compact topological group $G$ is called Van der Waerden if each homomorphism $h:G\to K$ to a compact topological group is continuous. By a classical result of Van der Waerden (1933) the groups $SO(...
- 40.2k
9
votes
1
answer
393
views
Meager subgroups of compact groups
Suppose we have an infinite compact (Hausdorff) group $G$, and a subgroup $H\leq G$ which is meagre.
Can $H$ always be covered by a countable family of nowhere dense sets $H_n$ such that $H_n^2$ is ...
- 28.8k