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Results tagged with descriptive-set-theory
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user 4832
Descriptive Set Theory is the study of definable subsets of Polish spaces, where definable is taken to mean from the Borel or projective hierarchies. Other topics include infinite games and determinacy, definable equivalence relations and Borel reductions between them, Polish groups, and effective descriptive set theory.
2
votes
Accepted
Is every convex subset of a Borel-linearly ordered space measurable?
I believe the answer is yes. (Please check this answer carefully, as this is rather outside my field. There may well be a much easier solution.)
In the paper "Borel Orderings" by Harrington, Marker …
- 29.8k
3
votes
Wanted: chain of nowhere dense subsets of the real line whose union is nonmeagre, or even co...
(Joel's answer appeared as I was typing this.)
I think the answer is no.
Suppose to the contrary there exists a nonmeager set $A \subset \mathbb{R}$ which is the union of some chain $\{K_i\}_{i \in …
- 29.8k
9
votes
Accepted
measurable linear functionals are also continuous on separable Banach spaces?
If "measurable" means "Borel" or even "Baire measurable", then this is true. It's a special case of a more general result that any Borel homomorphism of Polish groups is continuous. See for instance …
- 29.8k
2
votes
Accepted
Existence of a map connecting two marginals of a product measure
No. The measures $p, \bar{p}$ could be almost anything; there's nothing in the given conditions that forces $\bar{p}$ to be a possible pushforward of $p$.
As an extreme example, take $X = \{0\}$ to …
- 29.8k
3
votes
Does every separated measurable space embed into a power of $\{0,1\}$?
As in Joseph Van Name's comment, let $X$ be an uncountable set and $\Sigma$ the $\sigma$-algebra of all countable and cocountable sets. Then $(X,\Sigma)$ is separated.
First note that $(X,\Sigma)$ …
- 29.8k
4
votes
A non-Borel union of unit half-open squares
(This addresses a misinterpretation of the question, where $p$ can be chosen. I'll try to fix it.)
This seems too easy, so maybe I've misunderstood the question, but: let $L$ be the diagonal line $\ …
- 29.8k
4
votes
Meager subspaces of a Banach space and weak-* convergence
Update: Here is a ZFC (probably even ZF+DC) counterexample for Q1. It's from probability and kind of indirect, maybe someone will be able to find something shorter.
Let $X = C_0([0,1])$, the space …
- 29.8k
1
vote
Accepted
Meager subspaces of a Banach space and weak-* convergence
The answer to Q2 is No.
I am grateful to Damian Sobota for drawing my attention to the following paper:
Darst, R. B.
On a theorem of Nikodym with applications to weak convergence and von Neuman …
- 29.8k
5
votes
Accepted
A question about Borel sets on the unit interval
Unless I'm missing something, your measures $d\phi$ are precisely the atomless finite Borel measures (equivalently, Baire measures) on $[0,1]$. Then your condition on $A$ is that it is universally m …
- 29.8k
20
votes
Accepted
Is there a Borel subset of $ \mathbb{R}^{2} $, with finite vertical cross-sections, whose pr...
No, no such set exists. This is a special case of the Lusin–Novikov theorem; see e.g. Kechris, Classical Descriptive Set Theory, Theorem 18.10.
In general, let $X,Y$ be standard Borel spaces, and s …
- 29.8k