All Questions
9 questions
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
22
votes
2
answers
1k
views
Gently changing measure
This question was asked and bountied on MSE without answer, so I'm porting it here:
There's an easy way to change the measure of a set of reals by moving to a larger universe: simply make $\mathbb{R}$...
- 21.2k
3
votes
1
answer
300
views
Measurably-isomorphic subsets of polish spaces and the continuum hypothesis
In Theorem 2.7 in the following notes, we seem to assume the following statement.
Let $(\Omega,\mathcal F)$ be a Polish space, and $A\in\mathcal F$ an uncountable set. Then there exists a bijection ...
- 1,247
7
votes
2
answers
410
views
Characterizations of infinite compact Abelian groups and probability spaces based on the forcing notion they give
Let $G$ be an infinite compact Abelian group with the collection $\mathcal{B}$ of Borel subsets of $G$, and $m$ the (unique) normalized Haar measure on $\mathcal{B}$. This gives a natural forcing ...
- 32.1k
5
votes
2
answers
387
views
A variant of Freiling's Axiom of Symmetry and a weak form of the Continuum Hypothesis in models where all sets of reals are Lebesgue measurable
Consider the following variant of Freiling's Axiom of Symmetry, $\mathsf{AS}$, which will be denoted $A_{< 2^{\aleph_0}}$:
given any function $f$ from $\mathbb{R}$ into the families of of subsets ...
- 6,132
6
votes
2
answers
796
views
The First Failure of GCH in Large Cardinals Smaller than Measurables
A well known theorem by Scott says:
If $\kappa$ is a measurable cardinal and $\mu$ a normal measure on it and $\mu (\lbrace\lambda\in\kappa~|~2^{\lambda}=\lambda^{+}\rbrace)=1$ then $2^{\kappa}=\...
user42090
2
votes
1
answer
431
views
Are measures of a measurable cardinal measurable? (Edited and Updated Version)
Update: Regarding to Prof. Hamkins's guidance I restricted the questions to the "normal" measures to avoid trivial answers.
Definition: Let $\kappa$ be a measurable cardinal. Define:
$\mathbb{M}_{\...
user36136
12
votes
1
answer
744
views
Can we change the Lebesgue measure by forcing?
Suppose $M$ is a model of ZFC, and $\mu^M$ is the Lebesgue measure on $\mathbb R^M$ such that $\mu^M(\mathbb R^M)=1$. It is known that if $r$ is a Cohen real over $M$ and $N=M[r]$ then $\mu^N(\mathbb ...
- 39.7k
8
votes
1
answer
969
views
Probabilities independent of ZFC?
Hi guys,
is it possible to change the probability of an event via forcing? More precisely, is there an innocent looking question on the probability of "something" whose answer is independent of ZFC?
...
- 165