All Questions
34 questions
3
votes
2
answers
190
views
An uncountable measurable subset of $\Bbb R$ containing no nonempty perfect set
$\newcommand\R{\Bbb R}$Assuming the axiom of choice, is there an uncountable Lebesgue-measurable subset $S$ of $\R$ that contains no nonempty perfect set?
Of course, such a set $S$, if it exists, ...
- 128k
3
votes
0
answers
117
views
Which sigma-ideals in a sigma-algebra are contained in an ideal of null sets?
Let $X$ be a Polish space and $\mathcal{B}(X)$ be the $\sigma$-algebra of Borel subsets of $X$. Given a Borel probability measure $\mu$ on $X$, we write $\mathcal{N}(\mu) := \{ B \in \mathcal{B}(X) : \...
6
votes
1
answer
290
views
Subset of the reals with zero inner measure and "full" outer measure in $\mathsf{ZF}+\mathsf{DC}$
Working in $\mathsf{ZF}+\mathsf{DC}$ (that is, we are allowed to use Dependent Choice but not full choice), suppose that there exists a non-measurable subset of the unit interval $[0,1]$ (just non-...
25
votes
2
answers
2k
views
Writing a function on $\mathbb{R}$ as a sum of two injections
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. It is well-known that, using transfinite recursion with a well-ordering of $\mathbb{R}$, one can construct two injective functions $g,h: \...
- 4,265
5
votes
0
answers
216
views
Applications of Baire's theorem on functions of first class
I found the following theorem on page 32 of John Oxtoby's Measure and Category.
Theorem 7.3. If $f$ can be represented as the limit of an everywhere convergent sequence of continuous functions, then $...
- 297
4
votes
1
answer
351
views
$\sigma$-algebra generated by analytic sets
The Borel $\sigma$-algebra $\cal B$ on real numbers has many good properties. For instance, all continuous functions are $\cal B/\cal B$-measurable. On the other side, not only $\cal B$ is not ...
- 205
4
votes
1
answer
746
views
Can all uncountable (but small) families of sets with positive measure have an uncountable subfamily with an intersection of positive measure?
My general question was is it consistent that any uncountable family of less than $\mathrm{non}(\mathcal{N})$ sets, each with positive measure, has an uncountable subfamily $\mathcal{F}$ s.t. $\bigcap ...
- 135
1
vote
0
answers
155
views
$f:Y\to X$ 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 \...
4
votes
1
answer
718
views
Is every element of $\omega_1$ the rank of some Borel set?
It is well known that we can obtain the $\sigma$-algebra of Borel subsets of $2^{\omega}$ in the following way: Let $B_0$ be the collection of all open subsets of $2^{\omega}$. For $\alpha=\beta+1$, ...
- 1,799
5
votes
1
answer
314
views
Do Borel subsets of the plane with null sections have Borel projections?
This might be a very easy question, and it might be better for mathstackexchange in which case I apologize. I'm stuck on something an anonymous referee wrote to me about a paper of mine and I'm hoping ...
- 1,882
5
votes
1
answer
330
views
Function whose graph is a Borel relation
Suppose $f\colon\mathbb{R}^{\omega}\longrightarrow\mathbb{R}$ is a function such that
$$G(f):=\{(x,y)\in\mathbb{R}^{\omega}\times\mathbb{R}\mid f(x)=y\}$$
is a Borel set. Does it necessarily follow ...
- 1,799
1
vote
1
answer
132
views
Lambda system generated by a non-atomic collection
Consider a probability space $(X,\Sigma,P)$.
Let say that a collection $\mathcal{B}\subseteq\Sigma$ is non-atomic if for every $E\in\mathcal{B}$ and $\alpha\in(0,P(E))$ there exists $F\in\mathcal{B}$ ...
- 11
2
votes
1
answer
207
views
The measure of ideals generated by random reals
We assume that for every real $x$, $L[x]$ only contains countably many reals.
Given a set $X$ of reals, then $L$-ideal generated by $X$ is the smallest set $I$ of reals so that
For any reals $x\in ...
- 4,201
5
votes
1
answer
231
views
Can it be that universal measurability is preserved by projections?
I am currently discovering descriptive set theory—with much pleasure! It is something of a surprise to me that, while the Borel hierarchy is indexed by $\omega_1$, the projective hierarchy is only ...
- 284
11
votes
0
answers
381
views
Concerning Luzin-(N)-property
Definition: a function $f:\mathbb{R}\to \mathbb{R}$ has Luzin-(N)-Property if $f$ maps any null set to a null set.
By https://www.encyclopediaofmath.org/index.php/Luzin-N-property, it is known that ...
- 4,201
5
votes
1
answer
528
views
Base zero-dimensional spaces
Definition. A zero-dimensional topological space $X$ is called base zero-dimensional if for any base $\mathcal B$ of the topology that consists of closed-and-open sets in $X$, any open cover $\mathcal ...
- 41.8k
0
votes
1
answer
196
views
Support of a regular measure Reg
Let $K$ be compact, Hausdorff space but not necessarily metrizable. Let $\mathfrak{M}$ be the Borel $\sigma$ field over $K$ and $\mu$ be a positive Regular Borel measure on $K$. Let $S$ be a subset of ...
- 521
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
3
votes
1
answer
300
views
Measurably-isomorphic subsets of polish spaces and the continuum hypothesis
In Theorem 2.7 in the following notes, we seem to assume the following statement.
Let $(\Omega,\mathcal F)$ be a Polish space, and $A\in\mathcal F$ an uncountable set. Then there exists a bijection ...
- 1,247
4
votes
0
answers
195
views
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
6
votes
0
answers
180
views
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 ...
- 41.8k
5
votes
0
answers
138
views
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 ...
- 41.8k
11
votes
2
answers
483
views
The "strong" measure number
Beyond measure zero we have yet another measure-y notion of smallness: strong measure zero. A set $S\subseteq\mathbb{R}$ is strong measure zero if, for any $f:\mathbb{N}\rightarrow\mathbb{R}_{>0}$, ...
- 21.2k
12
votes
1
answer
316
views
A reference to a theorem on the equivalence of ideals of measure zero in the Cantor cube
I am looking for a reference of the following (true) fact:
Theorem. For any two continuous strictly positive Borel probability measures $\mu,\lambda$ on the Cantor cube $2^\omega$ there exists a ...
- 41.8k
13
votes
3
answers
820
views
Is there a Borel subset of $ \mathbb{R}^{2} $, with finite vertical cross-sections, whose projection onto the first component is non-Borel?
This question is related to another one that I asked two days ago.
Question. Does there exist a Borel subset $ M $ of $ \mathbb{R}^{2} $ with
the following two properties?
The ...
- 1,396
13
votes
1
answer
751
views
Is there a class of mathematical structures with non-isomorphic natural representations as a standard Borel space?
Background. The field of Borel equivalence relation theory
provides a robust, unifying theory that organizes most of the
classification problems of classical mathematics into a hierarchy,
allowing us ...
- 236k
3
votes
0
answers
689
views
"Nicely" strong measure zero sets
This question is essentially an expanded version of the unanswered half of Two strengthenings of "strong measure zero".
A set $X$ of reals is strong measure zero if, for any $f: \omega\...
- 21.2k
18
votes
1
answer
772
views
Two strengthenings of "strong measure zero"
A set $X\subseteq\mathbb{R}$ is strong measure zero if, for every sequence $(\epsilon_i)_{i\in\mathbb{N}}$ of positive reals, there is a sequence $(I_i)_{i\in\mathbb{N}}$ of open intervals covering $X$...
- 21.2k
2
votes
1
answer
397
views
From universal measurability to measurability
Let $(\Omega,\Sigma)$ be a measurable space and $K$ be a compact
metrizable space endowed with its Borel $\sigma$-algebra
$\mathcal{B}(K)$. Let $A\subseteq\Omega\times K$ be universally
...
11
votes
1
answer
799
views
Restrictions of null/meager ideal
Let I denote the null (resp. meager) ideal on reals. Is it consistent that for any pair of non null (resp. meager) sets A and B, there is a null (resp. meager) preserving bijection between A and B? In ...
- 9,631
13
votes
1
answer
1k
views
Is every $\sigma$-algebra of sets *abstractly* the Borel algebra of a topology on perhaps some other set?
Is every sigma-algebra the Borel algebra of a topology?
inspires the present question which asks for less.
Question: Given a $\sigma$-algebra $\mathcal A$ on a set $X$, does there exist a topology $\...
- 17.6k
6
votes
2
answers
392
views
Are Vitali-type nonmeasurable sets determinate?
Here, by a Vitali set, I mean the following. Call $f_1,f_2:\omega\rightarrow 2$ tail-equivalent if {$n| f_1(n)\not=f_2(n)$}$<\infty$. Vitali sets (existence via AC) contain one such $f$ from ...
- 17.6k
92
votes
3
answers
14k
views
Is every sigma-algebra the Borel algebra of a topology?
This question arises from the excellent question posed on math.SE
by Salvo Tringali, namely, Correspondence
between Borel algebras and topology.
Since the question was not answered there after some ...
- 236k
75
votes
4
answers
24k
views
Non-Borel sets without axiom of choice
This is a simple doubt of mine about the basics of measure theory, which should be easy for the logicians to answer. The example I know of non Borel sets would be a Hamel basis, which needs axiom of ...
- 7,442