All Questions
Tagged with measure-theory descriptive-set-theory
117 questions
3
votes
2
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180
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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
1
answer
69
views
How irregular can the set of points of non-differentiability for an L1 function's primitive F get, before the FTC fails?
A Fundamental Theorem of Calculus for Lebesgue Integration, J. J. Koliha begins with the passage
Lebesgue proved a number of remarkable results on the relation between integration and differentiation....
- 831
3
votes
0
answers
76
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\...
- 627
1
vote
1
answer
208
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Function $g:\mathbb{R}\to \mathbb{R}^n$ such that $g(\sum_{i=1}^nx_i)=(x_1,\dotsc,x_n)$ a.e
Is there a measurable function $g:\mathbb{R}\to \mathbb{R}^n$ such that $g(\sum_{i=1}^nx_i)=(x_1,\dotsc,x_n)$ a.e.?
Due to the papers [1], [2], and [3] I'm obtaining a result that I think it's false. ...
- 627
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) : \...
5
votes
0
answers
95
views
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, ...
- 291
4
votes
1
answer
311
views
If $f=h\circ g$, then there's a measurable function $\tilde h$ such that $f=\tilde h\circ g$
Let $X,Y,Z$ be three standard measurable spaces and $f:X\to Z$ and $g:X\to Y$ two measurable functions. Suppose that there's a function $h:Y\to Z$ such that $f=h\circ g$. How can I show that there's a ...
- 627
10
votes
1
answer
258
views
Sufficient condition for the graph of a measurable map to be measurable
Let $f:X \to Y$ be measurable map between measurable spaces w.r.t. to their corresponding $\sigma$-algebras $\Sigma_X$ and $\Sigma_Y$, resp.
If $(X,\Sigma_X)$ is a standard Borel space can we always ...
- 285
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
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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
-1
votes
1
answer
132
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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 ...
- 297
2
votes
1
answer
133
views
Can convergence in distribution necessarily be realised by almost-sure convergence?
Let $X$ be a Polish space. Let $(\mu_n)_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures $\mu_n$ on $X$ such that $\mu_n \to \mu_\infty$ weakly as $n \to \infty$. For each ...
- 708
2
votes
1
answer
370
views
Relationship between Baire sigma algebra and Borel sigma algebra of an uncountable product
I've been trying to understand various questions to do with sigma algebras on uncountable product spaces.
Let $T$ be an uncountable set and for each $t \in T$, let $\Omega_t$ be a topological space. ...
- 1,179
3
votes
0
answers
181
views
Which is the smallest $\sigma$-algebra that contains all analytic sets?
Let $X$ be a polish space. Is the smallest $\sigma$-Algebra containing all analytic sets of $X$ (i.e. all subsets $A \subset X$ which are the continuous image of a polish space) the $\sigma$-algebra ...
- 243
1
vote
0
answers
65
views
Is the projection of an universally measurable set again universally measurable?
Let $(X,\mathcal{A})$ be a measurable space and $(Y,\mathcal{B}(Y))$ be a polish space together with the Borel-$\sigma$-Algebra.
There is a Theorem that states: The projection $\pi_X(B)$ of every ...
- 243
4
votes
1
answer
166
views
Is the set of clopen subsets Borel in the Effros Borel space?
Let $X$ be a Polish space and $\mathcal{F}(X)$ the set of closed subsets of $X$ endowed with the Effros Borel structure, generated by sets of the form $\{F\in \mathcal{F}(X):F\cap U\neq \emptyset\}$, ...
- 3,115
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
1
vote
0
answers
83
views
Existence of a stronger notion of perfect measures
Let $\mathcal{X}$ be a measurable space with its $\sigma$-algebra $\mathcal{B}_\mathcal{X}$ and let $\mathbb{R}$ be the real numbers endowed with its Borel $\sigma$-algebra $\mathcal{B}_\mathbb{R}$.
...
- 285
6
votes
2
answers
288
views
Atoms for Markov kernels
Let $X$ and $Y$ be standard Borel measurable spaces. A Markov kernel $f : X \rightsquigarrow Y$ is a map $f(-|-) : \Sigma_Y \times X \to [0,1]$ such that:
$f(-|x)$ is a probability measure on $Y$ for ...
- 6,406
1
vote
1
answer
120
views
How to characterize the Borel sets of product between finite and uncountable space?
Consider the product space $Z=X\times Y$, where $X$ is a finite set with discrete topology and $Y$ is an uncountable compact subset of $\mathbb{R}^n$ with the usual subspace topology. Denote with $\...
- 11
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
1
vote
0
answers
64
views
Prescribed class of measurable sets
Let $X\neq\emptyset$ and let $\mu:P(X)\to[0,\infty]$ be an outer measure. Recall that, a set $A\subseteq X$ is $\mu$-measurable if
$$
\mu(B)=\mu(A\cap B)+\mu(B\setminus A), \text{ for all }B\subseteq ...
- 895
2
votes
1
answer
852
views
The Borel sigma-algebra of a product of two topological spaces
The following problem arose while trying to justify some "known results" in abstract harmonic analysis on noncommutative groups, for which I couldn't find explicit statements in the ...
- 25.8k
3
votes
1
answer
132
views
Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of Baire for every $F$?
Let $F\subset \Bbb{R}$ intersect every closed uncountable subsets of $\Bbb{R}$.
Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of ...
- 307
1
vote
0
answers
155
views
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
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
5
votes
1
answer
254
views
Is the topology of weak+Hausdorff convergence Polish?
Let $X$ be a compact metric space, $P_X$ the set of Borel probability measures on $X$, and $K_X$ the set of non-empty closed subsets of $X$. I will define the "topology of weak+Hausdorff ...
- 708
2
votes
1
answer
181
views
Non-analytically measurable set in $\Delta^1_2$
I'm wondering if there is some reference you may know that gives an explicit set which is not analytically-measurable (i.e., not in the sigma-algebra generated by $\Sigma^1_1$), but which is in $\...
- 119
2
votes
1
answer
148
views
Borel $\sigma$-algebras on paths of bounded variation
Let $(C, \|\cdot\|)$ be the Banach space of continuous paths $x: [0,1]\rightarrow\mathbb{R}^d$ starting at zero with sup-norm $\|\cdot\|$.
Let further $B\subset C$ be the subspace of $0$-started ...
- 463
1
vote
1
answer
183
views
Topological analog of the Lusin-N property
$A\subset \Bbb{R}$ is meager if $A$ can be expressed as a countable union of nowhere dense sets.
Let $f:[a, b]\to \Bbb{R}$ is absolutely continuous, i.e., for every $\epsilon>0$, there exists $\...
- 307
10
votes
0
answers
272
views
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
6
votes
1
answer
353
views
A strong Borel selection theorem for equivalence relations
In Kechris' book "Classical Descriptive Set Theory" there is the following theorem (12.16):
Let $X$ be a Polish space and $E$ an equivalence relation such that every equivalence class is ...
- 365
2
votes
0
answers
139
views
Are there any measurable spaces of functions
I am approaching this question from a probability perspective, and am hoping for some kind of framework to help understand all of this. I believe I may have even asked a similar question on here in ...
- 213
3
votes
0
answers
299
views
Universally measurable but not analytically measurable set
is there a concrete example (or even a non-concrete example) of a universally measurable set which is not analytically measurable (i.e., not in the sigma-algebra generated by the analytic sets)?
- 119
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
1
vote
2
answers
137
views
Locally compact Polish groups acting on standard Lebesgue spaces
If $G$ is a countable discrete group, then one can consider the Bernoulli shift $2^G$. $G$ acts on $2^G$ via shift, and letting $\mu$ be the product of the $(1/2, 1/2)$-measure in each coordinate, ...
- 369
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
0
votes
0
answers
185
views
Is the co-projection of the projection of a Borel set Lebesgue measurable?
Reading the classical book of Kechris "Classical descriptive set theory", I found the following facts
The projection of a Lebesgue measurable set need not be Lebesgue measurable, but the projection ...
- 21
2
votes
0
answers
261
views
Reference for Borel $\sigma$-algebra of topology of convergence in probability
I'm pretty sure I can prove the "Theorem" given further below (without very much difficulty), but it seems way too basic not to have been noticed before.
So I'm wondering if there are any papers/...
- 708
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
9
votes
0
answers
256
views
Is the inverse of a measurably parametrised family of bijections between standard Borel spaces measurably parametrised?
It is known that a measurable bijection $f \colon [0,1] \to [0,1]$ has a measurable inverse. (Here, all measurability is simply with respect to the Borel $\sigma$-algebra of $[0,1]$.)
Now fix an ...
- 708
5
votes
1
answer
310
views
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 ...
- 9,330
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
3
votes
0
answers
92
views
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
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