All Questions
5
questions
1
vote
0
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71
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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 ...
4
votes
1
answer
218
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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
vote
1
answer
53
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 ...
6
votes
0
answers
127
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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
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 ...