Tagged Questions

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Representative elements in the symmetric difference metric

The symmetric difference is a natural way to quantify the distance between measurable sets: $$d(S,T)=measure([S\setminus T]\cup[T\setminus S])$$ This is a pseudo-metric because there may be ...
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$\sigma$-field and Uncountable ordinal

I have been trying to get my head around this question. Any help greatly appreciated. Let $\mathscr{C}$ be any class of subsets $\Omega$ with $\emptyset,\Omega\in \mathscr{C}$. Define ...
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Connectedness of parts used in the Banach–Tarski paradox

A quote from the Wikipedia article "Axiom of choice": One example is the Banach–Tarski paradox which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ? ...
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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 ...
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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 ...
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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 ...