# Questions tagged [borel-sets]

For questions about Borel sets. Please, add also other tags indicating the area, e.g., (measure-theory), (general-topology), (descriptive-set-theory), etc.

279 questions
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### Borel action of $GL_n(\Bbb{Q})$ on $S(\Bbb{Q}^n)$

I'm reading a paper on torsion-free abelian groups and I'm trying to work out all the details. I know that the space of all torsion-free abelian groups of rank $1\le r\le n$, where $n$ is a positive ...
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### Does every uncountable Borel subset of $\mathbb R$ contains a perfect subset？

This question came from (London Mathematical Society Student Texts) Krzysztof Ciesielski-Set Theory for the Working Mathematician-Cambridge University Press. Chapter 6.2 Exercise 5. I have thought ...
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### Continuous then measurable?

I have this proposition: Prop. Every continuous functions $f:\mathbb{R}^n \to \mathbb{R}$ is $\mathcal{B^n} - \mathcal{B}$ measurable. I assume here $\mathbb{R}, \mathbb{R}^n$ count with their ...
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### Proof borel sets are measurable sets [closed]

Proof that : if $B$ is Borel Set, then $B$ measurable set. I know the definition of measureable sets, for all $A\subseteq \mathbb{R}$, $m^*(A)=m^*(A\cap E)+m^*(A\cap E^C)$, which $m*$ denote the ...
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### Is there such a term as a “Borel measurable set”?

Not sure if this is the right place to post such a rookie question, but I'd appreciate some quick clarification. Is there such a term as a "Borel measurable set"? I've seen it used all over the place ...
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### Building a Random Element From a Distribution

Suppose that $(X,\mathscr{B}(X))$ is a standard Borel space, and let $\pi$ be a Borel-probability measure on $X$. Is there a way to construct an $X$-valued random element, with distribution $\pi$?
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### Quotient Borel structure

I'm reading "Ergodic theory and Semisimple Lie groups" by Zimmer and (are p. $10$) the author states that ($G$ is locally compact and second countable): Definition $(2.1.9)$ Let $S$ be a Borel $G$...
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