All Questions
5
questions
1
vote
0
answers
71
views
Lebesgue measurability
For what $n$ it is (in)consistent that all $\Sigma^1_n/\Pi^1_n$ sets are Lebesgue measurable ?
I remember that there is a result that if all $\Sigma^1_3$ are Lebesgue measurable then
$\omega_1$ is ...
- 3,978
4
votes
1
answer
218
views
Continuum many reals with pairwise irrational difference
In "Problems and Theorems in Classical Set Theory" by Péter Komjáth and Vilmos Totik, in the Solutions to Chapter 30, they claim: "It is easy to give continuum many reals with pairwise ...
- 1,647
1
vote
1
answer
52
views
Is the intersection of a uncountable real numbers subset with the complemetary of a countable subset uncountable? [duplicate]
Let be $E,F\subset \Bbb{R}$ two subsets such that $E$ is uncountable and $F^c$ is countable. Is $E\cap F$ uncountable?
I guess it is true, but I am not sure since I don't see a way in order to prove ...
- 733
6
votes
0
answers
126
views
A 'measure' on $\mathcal{P}(\mathbb{R})$
Question: Is there function $\mu : \mathcal{P}(\mathbb{R}) \to [0, \infty]$ with the following properties:
$\mu$ is countably additive. (on disjoint sets)
$\mu((a, b])) = b-a$, i.e., it extends the ...
- 3,623
3
votes
1
answer
216
views
Can you prove that $\Bbb R$ is uncountable using the Lebesgue measure?
I have been studying measure theory from the ground up, and am quite excited by the seeming power it holds. I thought of this last evening, and I wish to ask if the following proof of uncountability ...
- 42.5k