All Questions
Tagged with axiom-of-choice measure-theory
23
questions
-6
votes
1
answer
124
views
Actual infinitesimals for solving Vitali paradox
Has anyone tried to use actual infinitesimals to solve paradoxes about non-measurability? In Vitali paradox, for example, they divide a set with measure 1 into $\infty$ subsets of zero measure and ...
- 111
3
votes
0
answers
190
views
Reverse-mathematical strength of Banach-Tarski
What is the reverse mathematical strength of the Banach-Tarski paradox?
The usual proof of Banach-Tarski should carry out in $\mathrm{ZF}+\mathrm{AC}_\kappa$, where $\kappa$ is the supremum of the ...
- 1,696
5
votes
0
answers
272
views
Completeness of the space $L^p$ and the Axiom of Countable Choice
I am thinking about the proof that the usual $L^p$ spaces are complete.
So, let $(X,\mathcal{F},\mu)$ be a measure space and let
$p\in[1,+\infty)$.
Important: by a measure I mean a nonnegative $\sigma$...
- 925
5
votes
0
answers
236
views
Polish transversals
A subset of $X$ an indecomposable continuum $Y$ is called a composant transversal if $X$ has exactly one point from each composant of $Y$.
So a continuum has a composant transversal precisely when ...
- 3,105
6
votes
1
answer
394
views
Measure of rational hyperplanes of $\mathbb{R}$
Let's view $\mathbb{R}$ as a vector space over $\mathbb{Q}$, and pick some basis $(v_\alpha)_{0 \leq \alpha < \mathfrak{c}}$ of it. We can then consider the subspace $L$ spanned by $(v_\alpha)_{0 &...
- 4,501
2
votes
0
answers
1k
views
The Rise and Fall of Dictators & How it Depends on Our Choice
This question is loosely inspired by the following paper of Shelah on Arrow property in which he answered a question of Gil Kalai affirmatively.
Shelah, Saharon, On the Arrow property. Adv. in Appl. ...
6
votes
1
answer
3k
views
Measurable functions with non measurable image
I am just curious about examples of measurable functions $f:[0,1]\to[0,1]$ such that $f[0,1]$ is not measurable.
This is motivated by the question Is measure preserving function almost surjective?, ...
- 1,805
7
votes
1
answer
790
views
author of a paradoxical decomposition of the interval
I am looking for the original author and the date of publication of the following result.
Theorem
There exist subsets $E_i\subset [0,1)$, $i\in {\bf Z}$, pairwise disjoints and real numbers $a_i$ ...
- 18.6k
3
votes
1
answer
311
views
Can a weaker version of the Hausdorff paradox be proved without AC?
The Hausdorff paradox is an incredibly counter-intuitive consequence of the axiom of choice; it is also important for demonstrating the non-existence, under AC, of a rotation-invariant measure on the ...
- 1,086
66
votes
9
answers
14k
views
Axiom of choice, Banach-Tarski and reality
The following is not a proper mathematical question but more of a metamathematical one. I hope it is nonetheless appropriate for this site.
One of the non-obvious consequences of the axiom of choice ...
- 10.3k
8
votes
2
answers
554
views
Darboux property of non-atomic sigma-additive nonnegative measures equivalent to the AC?
A result commonly, and probably erroneously, attributed to W. Sierpiński is that every non-atomic, countably additive, nonnegative measure $\mu: \Sigma \to \bf R$, where $\Sigma$ is a sigma-algebra on ...
- 9,675
4
votes
1
answer
2k
views
Does Borel's proof for existence of normal numbers make an essential use of axiom of choice?
A normal number is a real number whose infinite sequence of digits in every base $b$ is distributed uniformly in the sense that each of the $b$ digit values has the same natural density $\frac{1}{b}$, ...
8
votes
0
answers
535
views
A Banach-Tarski game
This is partially inspired by the question https://math.stackexchange.com/questions/1383397/cutting-a-banach-tarski-cake, which I find intriguing if unclearly written.
A paradoxical family of subsets ...
- 20.7k
11
votes
2
answers
1k
views
Is sigma-additivity of Lebesgue measure deducible from ZF?
Is sigma-additivity (countable additivity) of Lebesgue measure (say on measurable subsets of the real line) deducible from the Zermelo-Fraenkel set theory (without the axiom of choice)?
Note 1. ...
- 15.8k
5
votes
2
answers
2k
views
Are all models of ZF + DC + "All set of reals are lebesgue measurable" also models of CH? [duplicate]
Possible Duplicate:
Lebesgue Measurability and Weak CH
I have studied a little set theory and I found that Solovay constructed a model of ZF+DC+"All set of reals are Lebesgue measurable" and I ...
