All Questions
11 questions
5
votes
3
answers
261
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, ...
- 13.4k
11
votes
1
answer
950
views
Uniformization/measurable selection theorems
Let $X,Y$ be measurable spaces and $F\subseteq X\times Y$. We say that $f:X\to Y$ is a uniformization map for $F$ if $(x,f(x))\in F$ for each $x\in \pi_X(F)$ where $\pi_X$ is the left projection map. ...
- 660
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 ...
- 285
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 ...
- 5,474
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
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
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 ...
- 18.5k
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
5
votes
1
answer
767
views
Hausdorff approximating measures and Borel sets
Suppose $ 1 \leq m \leq n $ are integers and for each $ 0 < \delta < \infty $ let $\mathscr{H}^{m}_{\delta} $ be the size $ \delta $ approximating measure of the $ m $ dimensional Hausdorff ...
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- 1
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 ...
- 41.9k
9
votes
1
answer
569
views
strong measurability question
Let $X$ be a separable Banach space and $\mathcal L$ the collection of bounded linear operators on $X$. The strong operator topology has the sub-basis $\{B_{x,y,\epsilon}\colon x,y\in X,\epsilon>0\}...
- 10.1k