# Tagged Questions

**15**

votes

**2**answers

347 views

### construction of nonmeasurable sets

I have a history question for which I've had trouble finding a good answer.
The common story about nonmeasurable sets is that Vitali showed that one existed using the Axiom of Choice, and Lebesgue et ...

**0**

votes

**0**answers

130 views

### A not defined notion in Friedman's article about Generalized Fubini's Theorem

I intend to study Friedman's article, A Consistent Fubini-Tonelli Theorem for Nonmeasurable Functions (http://projecteuclid.org/download/pdf_1/euclid.ijm/1256047607). I think since I had a modern ...

**5**

votes

**2**answers

262 views

### non-Borel set which intersects every compact in a Borel set

I remember hearing some time ago that there is a locally compact Hausdorff space $X$ and a non-Borel subset $E$ which intersects every compact set in a Borel set. (This would contradict Lemma 13.9 of ...

**13**

votes

**0**answers

298 views

### A question about small sets of reals

In ZFC, does there exist an uncountable set of reals $A$ such that for every closed measure zero set of reals $B$, $ A + B = \{a+b : a \in A, b \in B\} \neq \mathbb{R}$?
This question is motivated ...

**4**

votes

**1**answer

160 views

### What does the set of cardinals admitting a k-additive measure look like?

Consider an infinite cardinal $\kappa$. Is it the case that the existence of a $\kappa$-additive measure on some infinite set implies the existence of such a measure on every infinite set of size ...

**3**

votes

**0**answers

177 views

### Strong measure zero sets and selection principles

A set of reals $X$ is strong measure zero if for any sequence of positive real numbers $ ( \epsilon_n ) _{n \in \omega }$ there is a sequence of open intervals $ ( a_n ) _{n \in \omega }$ which ...

**10**

votes

**2**answers

371 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. ...

**6**

votes

**2**answers

356 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**

votes

**1**answer

196 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:
...

**9**

votes

**0**answers

308 views

### Ultrafilter theorem and translation invariant measures

The usual Vitali construction of a non-Lebesgue measurable set generalizes to a proof that there are no (non-trivial) translation invariant measures on $\mathcal P\mathbb R$.
On the other hand, there ...

**23**

votes

**3**answers

781 views

### Does every set of reals contain a measure-zero set of the same cardinality? Does it contain a meager set of the same cardinality?

This question arises from an issue in my post on Ashutosh's excellent question on Restrictions of the null/meager ideal.
Question 1. Does every set of reals contain a measure-zero subset
of the same ...

**16**

votes

**2**answers

498 views

### Intersection of compact sets in the unit interval

Let $\mathscr K$ be an uncountable set such that every $K\in\mathscr K$ is a compact subset of $[0,1]$ with positive Lebesgue measure. Does it then follow that there exists an uncountable $\mathscr ...

**16**

votes

**1**answer

451 views

### How strong is “all sets are Lebesgue Measurable” in weaker contexts than ZF?

Famously, Solovay showed that, if $\textrm{ZFC}$ plus $\textrm{IC}$ (the existence of an inaccessible cardinal) is consistent, then so is $\textrm{ZF}$ plus $\textrm{DC}$ (dependent choice) plus ...

**6**

votes

**1**answer

500 views

### Do Measurable Cardinals Exist? (assuming ZFC)

In Appendix B of his Uniform Central Limit Theorems (1999), Dudley writes:
It is consistent with the usual axioms of set theory (including the axiom f choice) that there are no measurable ...

**4**

votes

**1**answer

212 views

### From universal measurability to measurability

Let $(\Omega,\Sigma)$ be a measurable space and $K$ be a compact
metrizable space endowed with its Borel $\sigma$-algebra
$\mathcal{B}(K)$. Let $A\subseteq\Omega\times K$ be universally
...

**10**

votes

**1**answer

399 views

### Restrictions of null/meager ideal

Let I denote the null (resp. meager) ideal on reals. Is it consistent that for any pair of non null (resp. meager) sets A and B, there is a null (resp. meager) preserving bijection between A and B? In ...

**5**

votes

**1**answer

331 views

### Is it consistent with ZFC that there is a translation-invariant extension of Lebesgue measure that assigns nonzero measure to some set of measure less than c?

It is consistent with ZFC (but not ZFC+CH, of course) that there is a subset $A$ of nonzero outer Lebesgue measure that has cardinality less than $c$. There will then be an extension of Lebesgue ...

**5**

votes

**1**answer

540 views

### A set of positive measure with cardinality less than that of the continuum?

Is it consistent with ZFC that there is a subset of $[0,1]$ whose cardinality is less than that of the continuum but which has positive Lebesgue measure?
Obviously not given CH. And, given ZFC, ...

**4**

votes

**2**answers

610 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 ...

**6**

votes

**0**answers

238 views

### Why has Sacks' “Measure-theoretic uniformity” not been more influential?

In the 1969 paper "Measure-theoretic uniformity in recursion theory
and set theory," Trans. Amer. Math. Soc. 142 1969 381â€“420, Sacks gave
a measure-theoretic approach to several results previously ...

**9**

votes

**4**answers

592 views

### Is there a maximum to the amount of disjoint non-measurable subsets of the unit interval with full outer measure?

This question arose a few years back when I was an assistant teacher on a course of basic (Lebesgue) measure theory, but I didn't find an answer or anyone able to solve the problem. The setting of the ...

**1**

vote

**1**answer

191 views

### Injective with Finite Discontinuities Mapping from $\mathbb R^n$ to $[0,1]$

Hi,
as a continuation to the fully answered question:
Injective&Intregrable Mapping from $\mathbb R^3$ to $\mathbb R$
Can one think of an injective $f:\mathbb R^n\rightarrow[0,1]$ that has only ...

**4**

votes

**2**answers

356 views

### Injective and Integrable Mapping from $\mathbb R^3$ to $\mathbb R$

Is there an injective and Riemann integrable map $f:\mathbb R^3\rightarrow\mathbb R$? (Of course such a map cannot be continuous.)

**2**

votes

**1**answer

404 views

### special extremally disconnected spaces with only finite isolated points

We Know that a cardinal $\kappa$ is measurable if there is a set $X$ with cardinal $\kappa$ and a {0,1}-measure $\mu: P(X) \rightarrow ${$0,1$} so that for all $x \in X$, $\mu(x)=0$ and $\mu(X)=1$. ...

**11**

votes

**2**answers

635 views

### Measuring big stuff

Often during informal discussion with colleagues, the following pattern emerges when we are stuck trying to prove a theorem about $x \in X$.
A: "let's assume this hypothesis $H$ on $x$"
B: "most ...

**8**

votes

**0**answers

467 views

### Is every $\sigma$-algebra of sets *abstractly* the Borel algebra of a topology on perhps some other set?

Is every sigma-algebra the Borel algebra of a topology?
inspires the present question which asks for less.
Question: Given a $\sigma$-algebra ${\cal A}$ on a set $X$, does there exist a topology ...

**7**

votes

**1**answer

542 views

### Intuition behind the diagonal intersection

Suppose that for all $\alpha<\kappa$ we have that $A_\alpha\subseteq\kappa$. We define the diagonal intersection to be $$\bigtriangleup_{\alpha<\kappa}A_\alpha = \left\lbrace\xi<\kappa\ ...

**4**

votes

**2**answers

537 views

### Finite measure on the power set

Let $X$ be an uncountable set, and let $\Omega$ be the power set of $X$, viewed as a $\sigma$-algebra. Does there exist a positive $\sigma$-additive measure of finite total mass on $(X, \Omega)$ such ...

**18**

votes

**4**answers

1k views

### Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?

One might say, "a random subset of $\mathbb{R}$ is not Lebesgue measurable" without really thinking about it. But if we unpack the standard definitions of all those terms (and work in ZFC), it's not ...

**6**

votes

**2**answers

277 views

### Are Vitali-type nonmeasurable sets determinate?

Here, by a Vitali set, I mean the following. Call $f_1,f_2:\omega\rightarrow 2$ tail-equivalent if {$n| f_1(n)\not=f_2(n)$}$<\infty$. Vitali sets (existence via AC) contain one such $f$ from ...

**9**

votes

**1**answer

448 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 ...

**44**

votes

**2**answers

3k views

### Is every sigma-algebra the Borel algebra of a topology?

This question arises from the excellent question posed on math.SE
by Salvo Tringali, namely, Correspondence
between Borel algebras and topology.
Since the question was not answered there after some ...

**3**

votes

**1**answer

521 views

### measure theory and continuum hypothesis

let's assume $\neg CH$, then there's a set $X$ such that $|\mathbb N|<|X|<|\mathbb R|$.
i'm wondering about the lebesgue measure of such set... is it even possible to measure it? would it be ...

**6**

votes

**2**answers

1k views

### Can we put a Probability Measure on every $\sigma$-Algebra?

The following question has puzzled me for some time:
Let $(\Omega,\Sigma)$ be a nonempty,
measurable space. Does there
necessarily exist a probability
measure $\mu:\Sigma\to[0,1]$?
If ...

**8**

votes

**2**answers

671 views

### Does there exist a subset of $\mathbb{R}^2$ which is “very small” and “very big” in the specified way?

Does there exist a set $M \subset \mathbb{R}^2$ which has the following two properties:
Forall $x \in \mathbb{R}$ the set $\{y \in \mathbb{R} \mid (x,y) \in M\}$ is countable.
Forall $y \in ...

**2**

votes

**2**answers

494 views

### Measure on $\omega_1$

Let $\mathcal{O}$ be the $\sigma$-algebra on $\omega_1$ generated by its countable subsets. Is there a ($\sigma$-additive) probability measure on $\mathcal{O}$ that is not concentrated on a countable ...

**8**

votes

**2**answers

813 views

### extensions of lebesgue measure

The Hahn-Banach theorem implies that Lebesgue measure can be extended give a "measure" on all subsets of [0,1], but this measure is only guaranteed to be finitely additive. It might magically turn ...

**18**

votes

**2**answers

2k views

### Can a Vitali set be Lebesgue measurable? (ZF)

Here is the definition of Lebesgue measure.
The standard proof that Vitali sets are not Lebesgue measurable uses countable additivity of Lebesgue measure, which is not a theorem of ZF. (In ...

**5**

votes

**2**answers

407 views

### Generalising Vitali Sets to uncountable dense subgroup selectors…

Does there exist an uncountable dense subgroup, $\Gamma$, of the additive group $\mathbb{R}$, such that every selector of the partition of $\mathbb{R}$ canonically associated with the equivalence ...

**6**

votes

**5**answers

877 views

### Is Lebesgue/Borel non-measurability actually caused by non-uniqueness?

In ZFC, every construction of a Lebesgue or Borel non-measurable set uses the axiom of choice. None of them that I've seen use choice to define a unique set, even though it's entirely possible to do ...

**1**

vote

**0**answers

122 views

### Follow up question on the measure of the difference between a partial selector and a selector…

This is a different question from my previous question Difference between a partial selector and a selector, however I am going to repeat the preamble...
In Kharazishvili's "Nonmeasurable Sets and ...

**0**

votes

**1**answer

137 views

### Difference between a partial selector and a selector…

In Kharazishvili's "Nonmeasurable Sets and Functions" there is the following theorem:
There exists a subset $X$ of $\mathbb{R}$ which is a Vitali set and a Bernstein set.
The proof is as ...

**7**

votes

**1**answer

800 views

### Universally measurable sets and weak topology

After I posted this question, a couple of months ago, and got from MO-users several
good hints, I think i'm ready, after some study, to ask another related question (or rather, to focus on the main ...

**1**

vote

**1**answer

1k views

### Lebesgue measure of the graph of a function [closed]

Let $f: R^n \rightarrow R^m$ be any function.
Will the graph of f always have Lebesgue measure zero ?
1) I could prove that this is true if f is continuous.
2) I suspect it is true if f is ...

**3**

votes

**1**answer

926 views

### $\Delta_{2}^{1}$-hard set?

Hello everybody!
I'm interested in $\Delta_{2}^{1}$ subsets of Polish spaces, i.e. those sets that are both $\Pi_{2}^{1}$ and $\Sigma_{2}^{1}$ in the boldface hierarchy of Polish spaces.
There is a ...

**8**

votes

**1**answer

672 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?
...

**1**

vote

**1**answer

517 views

### Probability Measures and Cardinality > c

Is it possible to place non-trivial probability measures on sets of cardinality strictly greater than the continuum -- in particular, on sets of cardinality 2^c? (Any references would be ...

**2**

votes

**2**answers

715 views

### measurability of integrated functions

Hello everybody,
DISCLAIMER: I'm not a mathematician, but a computer scientist, so I hope the question is not trivial (or perhaps I hope so, in order to get a definitive answer). Anyway it's not a ...

**22**

votes

**2**answers

693 views

### Codimension of Measurable Sets

I am currently teaching an advanced undergraduate analysis class, and the following question came up.
Intuition suggests that "most" subsets of $[0,1]$ are not Lebesgue measurable. However, the ...

**18**

votes

**1**answer

645 views

### A collection of intervals that can cover any measure zero set

This is a follow-up to this question (in fact, this is what originally motivated me to ask that one.)
Let's say that a sequence $\{s_i\}$ of positive reals covers a set $X\subset\mathbb R$ if there ...