12
votes
Accepted
Is $X\times X$ homeomorphic to $X$ for a space of probability measures?
The answer is yes, following from Klee's extension of Keller's theorem on homeomorphisms of infinite dimensional compact convex sets:
Klee, V. L., Some topological properties of convex sets, Trans. Am....
- 3,768
7
votes
Which self homeomorphisms preserve measure on a torus, apart from affine?
As a simple example, if you take any divergence-free vector field, the flow generated by this vector field is incompressible, so preserves measures.
In fact, there are a huge number of measure ...
- 6,001
6
votes
An uncountable measurable subset of $\Bbb R$ containing no nonempty perfect set
Yes, AC gives us a continuums-sized measure 0 set without the perfect set property.
For the construction of just any continuums-sized subset of $\mathbb{R}$, what matters is that $\mathbb{R}$ has ...
- 4,727
5
votes
An uncountable measurable subset of $\Bbb R$ containing no nonempty perfect set
Assuming the axiom of choice, there is a (may I say very natural) uncountable set of real numbers that is measure-zero with regard to any $\sigma$-finite, complete, regular measure that measures all ...
4
votes
Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine?
Take any homeomorphism $h$ of the unit circle (say $C$).
Composing $h$ with an appropriate rotation, we see that, without loss of generality, $1=1+0i$ is a fixed point of $h$.
Considering ...
- 128k
3
votes
Accepted
Comparing two different principles of premeasure-to-measure extension
$\newcommand\Om\Omega\newcommand\N{\Bbb N}\newcommand\R{\Bbb R}\newcommand\M{\mathscr M}\newcommand\A{\mathscr A}$The equality $\Om_N=\Om_C$ follows immediately from Theorem 1.4 and Remark 1.5 in this ...
- 128k
2
votes
Accepted
An uncountable measurable subset of $\Bbb R$ containing no nonempty perfect set
A set $B\subseteq\mathbb R$ is a Bernstein set if $B$ contains no nonempty perfect set while having nonempty intersection with every nonempty perfect set; in other words, if neither $B$ nor $\mathbb R\...
- 13.4k
2
votes
Counterexamples to differentiation under integral sign?
Here's an example that came up in practice.
Theorem: (Cauchy-Pompeiu)
Let $U \subset \mathbb{C}$ be a bounded open set with piecewise- $C^1$ boundary $\partial U$ oriented positively (see appendix B )...
- 833
2
votes
Accepted
How irregular can the set of points of non-differentiability for an L1 function's primitive F get, before the FTC fails?
I realized perhaps my question is equivalent to asking for a characterization of sets $S\subseteq [a,b]$ that can not support a singular continuous measure.
Actually, here we may assume that $S$ is ...
- 128k
2
votes
Accepted
Every tight $\tau$-additive finite measure is Radon
It seems that the part you're having trouble with is proving that the restriction of a bounded $\tau$-additive Borel measure to a Borel subset is $\tau$-additive (and you can follow the rest of ...
- 3,768
2
votes
Accepted
Existence of a "universal" measure-preserving transformation on the unit interval
As written, this has no chance. If $A$ isn't invariant, what does it mean to say $T|_A$ is isomorphic to $S$? Even if you restrict to invariant sets, you still have a problem with the positive measure ...
- 23.2k
1
vote
Accepted
Does convergence in probability of iid samples imply convergence in measure of the sampled functions?
Counterexample: $g_i(x)=x-1/2$ for all $i$ and all $x\in[0,1]$.
- 128k
1
vote
Accepted
Existence of asymptotic sequence in ergodic measure-preserving transformations
You can define $\phi_n=S_n(1_A)/\theta_n$ where $S_n(1_A)=\sum_{k=0}^{n-1}1_A\circ T^k$ and view $\phi_n$ as funcitons in $L^2(A,\mu)$.
The condition you mentioned gives that $\|\phi_n\|_2\leq \lambda$...
- 111
1
vote
Proof of the Dunford-Pettis theorem in the context of probability spaces
In some lecture notes here, pages 7, 8 and 9, a proof is given. The direction 2. $\Rightarrow $ 1. rests on extraction of a sub-sequence such that $\left(X_{n_k}\mathbf{1}_{\lvert X_{n_k}\vert \...
- 3,968
1
vote
Accepted
Measure on the places of $\bar{\mathbb Q}$
Allcock and Vaaler give a description of the natural measure $\lambda$ on the space of places of $\overline{\mathbb Q}$ in this paper, specifically in Section 4. As was mentioned in the comments, for ...
- 704
Only top scored, non community-wiki answers of a minimum length are eligible