# Tagged Questions

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### Limits of sets question

Suppose $A_n \rightarrow A$. Where convergence is defined in a usual way as: \begin{align} \limsup A_n=\bigcap_{n}\bigcup_{k \ge n} A_k=\liminf A_n=\bigcup_{n}\bigcap_{k \ge n} A_k \end{align} My ...
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### Summable enumerations of $\Bbb Q$

We say that a set of natural numbers $A$ is summable if $\sum_{n\in A}\frac1n$ is finite. It is not hard to see that $\{A\subseteq\Bbb N\mid A\text{ is summable}\}$ is an ideal on $\Bbb N$: Subsets ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $cov(null)$ 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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]: ...
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### 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 ...
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### 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 $X$ ...
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### 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) = ...
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### 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 ...
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### 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 ...
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### 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 $E$ ...
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### 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 $\mu: F \rightarrow [0,\infty)$ ...
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### 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 ...
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### 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...
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### 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 ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 , ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 $I$ ...
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 ...