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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.

For questions about Borel sets.

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808 questions
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I'm trying to prove the following proposition concerning the measurabilty of a set. Is my proof correct?

Proposition: Let $(X,\mathcal{A})$ be a measurable space, let $Y$ be a separable metrizable space, and let $f,g: X \to Y$ be measurable with respect to $\mathcal{A}-\mathcal{B}(Y)$ , where $\mathcal{B}(Y)$ denotes the Borel-$\sigma$-Algebra. Show…
user679342
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Intuition help: why are Borel sets important?

When you start reading measure theory, the first motivation presented is we want to find some nice sets we can measure. The solution is $\sigma$-algebras. Then we define a random variable as $X: \Omega \rightarrow \mathbb{R}$. Both $\Omega$ and…
Cinad
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Is A Borel set either a $G_\delta$ or a $F_\sigma$?

On my lecture notes I read the following statement: A Borel set can be either a countable union of closed sets ($F_\sigma$) or a countable intersection of open sets ($G_\delta$) Does it mean, for example, that given a set $A \in B(X)$ (where $X$ is…
Julian Vené
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Exclude the "closure under complementary" axiom from $\sigma$-algebra

I hope this is not answered already on stackexchange because I cannot quite search it using minimal number of words, but I have heard of the Borel Hierarchy and suspected that the axiom "closure under complementary " is why the construction of…
user305521
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1 answer

Closed Set - Closed Set = $F_{\sigma}$ set?

Let $A$ and $B$ be closed sets in $\mathbb{R}$. Is $A\setminus B$ an $F_{\sigma}$ set?
APR
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When do we take the size of Borel sets?

Borel sets are defined because we want a measurable set of sets, and we want them to have nice topological properties. .... Uhm.... but when do we actually take the size of Borel sets? When are they ever measured? In Probability Theory, we measure…
Cinad
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Is my understanding about Borel $\sigma$-algebra correct

Regarding the definition of Borel $\sigma$-algebra, let $\Omega=\mathbb{R}$, $\mathcal{O}=\{(a,b):-\infty < a \le b < \infty\}$, can I write Borel $\sigma$-algebra: $\sigma(\mathcal{O})$ as $\sigma(\mathcal{O}) = \{…
Ben
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Borel set and power set

A Borel set of $\mathbb R$ is equivalent to an interval. An element of $\mathscr P(\mathbb R)$ is also an interval. So, can we say a Borel set on $\mathbb R$ is a part of $\mathbb R$? More generally, what is the use of Borel sets of a set $X$ if we…
niobium
  • 1,177
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Contradiction of the Borel sigma-field when we think the set of irrational numbers.

I understand the sigma-field as not allowing uncountable union or intersection. In set of irrational numbers, it is the Borel set because it is a complement set of rational numbers. But if we think it as a union of the irrational numbers, it is a…
이선훈
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How do I prove that an increasing function is (Borel)-mesurable?

I need to show that an increasing function $f:\mathbb{R}\rightarrow \mathbb{R}$ is (borel)-mesurable proof From the lecture we know that since $B(\mathbb{R})=\langle Q(\mathbb{R})^\sigma\rangle$, it's enough to show that $\forall E\in…
user123234
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Is $(a, \infty]$ a Borel set?

Is interval $(a, \infty]$ a Borel set ? I hear that if $U$ is written as union, intersection, or subtraction of open sets, $U$ is called a Borel set. I wonder whether $(a, \infty]$ is a Borel set or not. $\cup_{n=1}^{\infty} (a, n]$ isn't equal to…
daㅤ
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open sets of $\mathbb{N}$

I would like to define a $\mathbb{N}$-valued random variable to define the $\sigma$-algebra generated by this random variable. In my course, we've define only $\sigma$-algebras on borelians of $\mathbb{R}$ ( by the set $\{ X^{-1}(A), A \in…
wainwain
  • 157
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Prove that this set is a borel set

I have difficulties proving the following : Let A be the set of all real x between [0 , 1], such that x has a decimal representation and each digit has infinite occurence. Now, what i do understand is, that a borel set is a set that can be…
bsvgu
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2 answers

what is difference between Borel algebra and Borel σ-algebra?

I am a university student, professor gave us definitions of Borel algebra and Borel σ-algebra. In Wikipedia they are the same see here. but professor defined Borel algebra as: let $\Omega$ be $[a,b)$ set on real line. Borel algebra on $\Omega$ is…
dato nefaridze
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image of homeomorphism Borel?

Let $X,Y$ be a Polish space, $A\subset X$ a Borel subset and $f:A \to B\subset Y$ a homeomorphism. Is $B$ then still Borel in $Y$ ?
user679342
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