3
votes
1answer
48 views

Constructively generating a sigma algebra

We have a collection $\mathcal{C}$ of sets (includes $\Omega)$ and would like to constructively generate the sigma algebra $\sigma(\mathcal{C})$. Would the following process work? Let ...
5
votes
1answer
144 views

How large are measurable cardinals of higher orders?

For each ordinal $\alpha$ define the notions of $\alpha$ - measurable cardinals and $\alpha$ - normal measures as follows: A measure $\mu$ on a measurable cardinal $\kappa$ is a $0$-normal measure ...
7
votes
0answers
75 views

Carlson's model and Sierpinski sets

Suppose that every countably generated sigma algebra extending the Borel sigma algebra on $[0, 1]$ admits a measure extending the Lebesgue measure (This is true in Carlson's model which is obtained by ...
1
vote
1answer
69 views

Banach Mazur game - Oxtoby - Thm 6.1

I have asked about this theorem before but found lately that I still don't fully understand its proof. Here are the rules of the game described. A closed interval in $\mathbb{R}$ denoted $I_0$ is ...
0
votes
1answer
51 views

Oxtoby - Thm 6.2 - There exists a strategy in which (A) wins iff $I_1 \cap B$ of first category

I am reading the book Measure and Category of Oxtoby. In Theorem 6.2 it is stated that There exists a strategy in which (A) wins iff $I_1 \cap B$ of first category As in the picture below. My ...
0
votes
2answers
120 views

$cov(null)$ in a Cohen model

Let $cov(null)$ be the minimum cardinality of a set of nullsets of $\mathbb{R}$ so that their union equals $\mathbb{R}$. In a model $M[G]$ obtained from a ctm $M$ with $M\models CH$ by adding ...
7
votes
1answer
112 views

Can Tarski's circle squaring problem be solved with measurable sets and/or without the Axiom of Choice?

Tarski asked whether a disk can be decomposed into finitely many pieces which can be rearranged into a square (necessarily of the same area by the failure of the Banach-Tarski paradox in two ...
7
votes
2answers
191 views

Does $\mathcal P ( \mathbb R ) \otimes \mathcal P ( \mathbb R ) = \mathcal P ( \mathbb R \times \mathbb R )$?

Obviously the answer's yes if I replace $\mathbb R$ by a countable set. If $\mathbb R$ is replaced by a set $X$ having a greater cardinality, then the diagonal $\{ (x,x) , \ x\in X\}$ is not in the ...
10
votes
3answers
293 views

Can sets of cardinality $\aleph_1$ have nonzero measure?

$\aleph_1$ is the cardinality of the countable ordinals. It is the least cardinal number greater than $\aleph_0$, and assuming the continuum hypothesis it's equal to $\mathfrak{c}$, the cardinality of ...
3
votes
2answers
98 views

Intersection of a directed family of large sets

Fix an $0\lt\varepsilon \lt 1$ (you may assume, if required, that $\varepsilon\lt \frac{1}{2}$ or, indeed, that $\varepsilon$ is sufficiently small for any argument you may wish to do). Let $G$ be a ...
1
vote
1answer
43 views

Problem understanding a statement in Oxtoby's “Measure and Category”, p.79. / Application of Zorn's lemma

Assuming we already "know" that any set $E$ of positive outer measure contains a family of $\mathfrak{c}=2^{\aleph_0}$ disjoint non-measurable sets. The statement I can't see is [2nd ed., p. 79]: ...
2
votes
2answers
139 views

Existence of minimal $\sigma$-algebra and transfinite induction

It is well-known that: Given a set $X$ and a collection $\cal S$ of subsets of $X$, there exists a $\sigma$-algebra $\cal B$ containing $\cal S$, such that $\cal B$ is the smallest ...
2
votes
1answer
70 views

existence of a subset that carries all measure of a chain

Suppose $m: \mathcal{P}(X) \to [0, 1]$ is a $\sigma$-additive measure on all subsets of $X$ and $\mathcal{L} = \{A_\alpha: \alpha < \beta\}$ is a well ordered by inclusion chain of subsets of $X$ ...
3
votes
2answers
102 views

Cardinality and Measurability

We can show that $\mathbb{R}$ and $\mathbb{R}^2$ or ($\mathbb{R}^n$) have same cardinality using the following one-to-one and onto mapping: Say x = (0.123456789....) Then f(x) = ...
1
vote
1answer
103 views

How many Borel conjectures are there

The following may be referred to as Borel conjecture: Every strong measure zero set of reals is countable. On the other hand Wikipedia refers to the following as the Borel conjecture: Let $M$ and ...
9
votes
1answer
239 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 ...
3
votes
1answer
102 views

Existence of a probability measure under which a subset of $[0,1]$ cannot be approximated by a Borel set

Is there some probability measure, $p$, on $\left(\left[0,1\right],2^{\left[0,1\right]}\right)$ relative to which some $D\subseteq\left[0,1\right]$ cannot be approximated by a Borel set, i.e. if $E$ ...
3
votes
1answer
81 views

Measurability of a function defined on a product measure space, and related to a measurable function

Let $ (X,\mu) $ be a standard measure space - so that we may assume that $X$ is the unit interval $[0,1]$ with the Borel $\sigma$-algebra. Consider $X \times X$ with the product measure $\mu \times ...
3
votes
2answers
326 views

Intersection of $\sigma$-algebras and set theory

Theorem: Given $\{E_{\alpha}\}_{\alpha \in \mathcal{A}}$, where each $E_\alpha$ is a $\sigma$-algebra on $X$. Then $E:=\bigcap_{\alpha \in \mathcal{A}}E_\alpha$ is a $\sigma$-algebra. Proof: Take ...
9
votes
0answers
157 views

Restrictions of null/meager ideal

Let I denote the null/meager ideal on reals. Is it consistent that for any pair of non null/meager sets A and B, there is a null/meager preserving bijection between A and B? In particular, is this ...
1
vote
1answer
122 views

A proof about $\sigma$-algebras via transfinite induction

This is a proofreading question. I was trying to help out on this question and in the course of that I encountered the following assertion: Let $(X, \mathcal A)$ and $(Y, \mathcal B)$ be ...
0
votes
0answers
132 views

Reference request: set theory of sigma algebras

I am studying Billingsley's Probability and Measure. The section on sigma-fields (Section 2) seems to demand set-theoretic reasoning beyond what I have been exposed to so far in undergraduate algebra ...
5
votes
2answers
232 views

What is a “linear set”

I'm reading "L'hypothèse du continu" by Sierpinski. He mentions many time "ensembles linéaires" or "linear sets" without defining this notion. Does anyone know what is the definition a such a set ? ...
6
votes
1answer
129 views

Given a model of ZF where $ \mathbb{R} $ is the countable union of countable sets, does every subset of $ \mathbb{R} $ have measure zero?

The question basically says it all. It is a well-known result that there exists a model $ \mathcal{M} $ of ZF with the property that $ \mathbb{R}^{\mathcal{M}} $ (here, $ \mathbb{R}^{\mathcal{M}} $ is ...
18
votes
1answer
545 views

What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?

What are some good intuitions for understanding Souslin's operation $\mathcal{A}$? Recall the definition: Let $S = \mathbb{N^{<N}} = \bigcup_{n = 1}^\infty \mathbb{N}^n$ be the set of ...
7
votes
1answer
287 views

Existence of non-atomic probability measure for given measure zero sets

Let $\Omega$ be a set and $\Sigma$ be a $\sigma$-algebra of subsets of $\Omega$. Let $N$ be a collection of measurable subsets of $\Sigma$. Question: What conditions on $\Sigma$ and $N$ guarantee ...
6
votes
3answers
796 views

Example for fintely additive but not countably additive probability measure

A probability measure defined on a sample space $\Omega$ has the following properties: For each $E \subset \Omega$, $0 \le P(E) \le 1$ $P(\Omega) = 1$ If $E_1$ and $E_2$ are disjoint subsets $P(E_1 ...
1
vote
2answers
645 views

Finitely but not countably additive set function

Let X be any countable! set and and let F be the cofinite set, i.e., $A \in F $ if A or $A^{c}$ is finite (this is an algebra). Then show that the set function $\mu: F \rightarrow [0,\infty)$ ...
20
votes
1answer
602 views

Universally measurable sets of $\mathbb{R}^2$

$$\text{Is }{{\cal B}(\mathbb{R}^2})^u={{\cal B}(\mathbb{R}})^u\times {{\cal B}(\mathbb{R}})^u\,?\tag1$$ Is the $\sigma$-algebra of universally measurable sets on $\mathbb{R}^2$ equal to the product ...
1
vote
2answers
377 views

Advantage of accepting non-measurable sets

What would be the advantage of accepting non-measurable sets? I personally feel that non-measurable sets only exist because of infamous Banach-Tarski paradox...
1
vote
2answers
141 views

Measure on Baire space

Inspired by the first parenthetical sentence of Joel's answer to this question, I have the following question: is there any useful notion of measurability in the Baire space $\omega^\omega$? Some ...
1
vote
1answer
229 views

Distribution Functions of Measures and Countable Sets

Let $\mu$ be a continuous probability measure on $[0,1]$. Then, the function $g:[0,1] \to [0,1]$ defined by $g(x) = \mu([0,x])$ is called the distribution function of $\mu$. I have proved that $g$ is ...
12
votes
1answer
201 views

Are sets constructed using only ZF measurable using ZFC?

Suppose $S$ is a subset of $\mathbb{R}$ which can be defined without using the axiom of choice, i.e. which can be proved to exist using only the axioms of ZF. Does it follow that $S$ is measurable? ...
3
votes
1answer
139 views

Measurable cardinal existence condition

We know that a cardinal $k > \omega$ is measurable if there is a measure function $\mu :2^k\mapsto \{0, 1\}$ that satisfies the following 3 conditions: 1.$<k$ -additive: for every set I of ...
9
votes
1answer
677 views

Can one construct a non-measurable set without Axiom of choice?

Is axiom of choice required to show the existence of non-measurable sets? Is there a Lebesgue non-measurable set that can be constructed without axiom of choice? Related question on MO says it is ...
6
votes
1answer
384 views

Existence of non-atomic probability measure

The Question Let $X$ be a set. Let $\mathcal{F}\subseteq P(X)$ be a $\sigma$-algebra. (Or, if it makes a difference, let $X$ be a topological space and $\mathcal{F}$ the Borel sets.) When can we ...
9
votes
3answers
950 views

Axiom of choice, non-measurable sets, countable unions

I have been looking through several mathoverflow posts, especially these ones http://mathoverflow.net/questions/32720/non-borel-sets-without-axiom-of-choice , ...
18
votes
1answer
3k views

Cardinality of Borel sigma algebra

It seems it's well known that if a sigma algebra is generated by countably many sets, then the cardinality of it is either finite or $c$ (the cardinality of continuum). But it seems hard to prove it, ...
15
votes
1answer
652 views

Medial Limit of Mokobodzki (case of Banach Limit)

A classical Banach limit is very usefull concept for me, but there is a problem with the integration and even with the measurability, this means for a sequence $(f_n)_{n\in \mathbb{N}}$ of measurable ...
7
votes
1answer
194 views

Fixed point: sets and measures

Let $X$ be a Borel space with a Borel measure $\mu$. Suppose $\xi: X\times X\to\mathbb R_{\geq 0}$ is a continuous function and put $s(x) = \{y\in X:\xi(x,y) = 0\}$. For any set $b\in\mathcal B(X)$ we ...
2
votes
0answers
160 views

Can Egoroff's Theorem be Strengthened for A Sequence of Smooth Functions?

I have posted this question on MO, and I'll raise the question here in a more concrete way. To simplify the situaton, we assume the measure space we are dealing with right now is a closed interval $I$ ...
3
votes
1answer
272 views

If you randomly choose a subset of the real line, what is the probability that it will be measurable?

Suppose that you are working with the axiom of choice. If you randomly choose a subset of the real line, what is the probability that it will be measurable?
14
votes
3answers
368 views

Locally non-enumerable dense subsets of R

Today after lunch I was hungry for math problems so I started begging for some at the department and finally someone threw me this: Can $\mathbb{R}$ be partitioned into two non-countable dense ...
4
votes
1answer
211 views

Does this make sense $\aleph_0+\aleph_1+\aleph_2$?

Let $\mathcal{A}$ denote the collection of all subsets A of an uncountable set $\Omega$ for which either A or $A^c$ are countable. Let $\mu(A)$ denote the cardinality of A. Define $\phi(A)$ equal to ...
5
votes
1answer
439 views

Banach Tarski Paradox

I know that Banach Tarski Paradox gives that "A three-dimensional Euclidean ball is equidecomposable with two copies of itself" and I have read that doubling the ball can be accomplished with five ...