Questions tagged [borel-sets]
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108 questions
8
votes
3
answers
846
views
A compactness property for Borel sets
Is the following generalised compactness property of Borel sets in a Polish space consistent with ZFC?
($*$) Let $\mathcal{B}$ be a family of $\aleph_1$-many Borel sets. If $\bigcap \mathcal{B} = \...
- 866
6
votes
10
answers
8k
views
Best introduction to probability spaces, convergence, spectral analysis
I'm not sure if this stuff all falls under what most would just term "probability", but I'm researching applied macroeconomics and need to get a handle on the following concepts ASAP:
probability ...
- 163
16
votes
5
answers
9k
views
A G-delta-sigma that is not F-sigma?
A subset of $\mathbb{R}^n$ is
$G_\delta$ if it is the intersection
of countably many open sets
$F_\sigma$ if it is the union of countably many closed sets
$G_{\delta\sigma}$ if it is the union
of ...
- 1,981
13
votes
6
answers
3k
views
Sets with equal positive measure in every interval
Hi,
I want to write a proof that relies on the fact that:
There are Borel Sets $A$ and $B$ contained in $\mathbb{R}$ such that
$A \cap B = \emptyset$ and $\lambda(A \cap (x,y)) = \lambda(B \cap (x,...
- 133
4
votes
2
answers
767
views
Possible subsets of reals that equal the set of continuity of a function
This should be an easy question, but I don't quite know how to approach it. It may be somewhat related to the concepts mentioned in the context of this past question, though it was motivated mainly by ...
- 7,320
19
votes
5
answers
10k
views
Projection of Borel set from $R^2$ to $R^1$
This should be easy to prove but I have no idea how to do it:
If $X \subseteq \mathbb{R}^2$ is borel then $f(X)$ is borel where $f(x,y) = x$
Thanks
Tobias
- 365
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
7
votes
4
answers
983
views
Higher-rank Borel sets
What are interesting, illustrative examples of Borel sets, situated in Borel hierarchy higher than $\Sigma^{0}_{2}$ /$\Pi^{0}_{2}$?
- 918