Linked Questions

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1answer
45 views

A question on Borel measurability

Let $X$ be a compact metric space. Given a map $x\mapsto E_x$ where $E_x\subset X$ is a Borel set in X. What can be said about the Borel measurability of the set $F_E:=\{(x,y)\in X\times X:y\in E_x\}$?...
34
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1answer
18k views

Lebesgue measurable set that is not a Borel measurable set

exact duplicate of Lebesgue measurable but not Borel measurable BUT! can you please translate Miguel's answer and expand it with a formal proof? I'm totally stuck... In short: Is there a Lebesgue ...
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0answers
42 views

Not all sets in $\mathbb{R}$ are Lebesgue measurable sets [duplicate]

Let $\mathcal{B}(\mathbb{R})$ be the Borel-sigma algebra on $\mathbb{R}$, and let $\mathcal{L}$ be the Lebesgue sigma-algebra which is a unique extension of $\mathcal{B}(\mathbb{R})$ with respect to ...
1
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1answer
432 views

How to prove that cardinality Borel $\sigma$-algebra equals the cardinality of $\mathbb R$? [duplicate]

My understanding at this point is that to assign a probability measure to a random variable defined on the real line, we need a Borel $\mathscr{B}$ sigma algebra, because otherwise we wouldn't be able ...
17
votes
1answer
476 views

Is there a set $A \subset [0,1]$ such that both $A$ and $[0,1] \setminus A$ intersect every positive-measure set?

Is there a set $A \subset [0,1]$ such that for every Borel set $B \subset [0,1]$ of positive Lebesgue measure, both $B \cap A$ and $B \setminus A$ are non-empty? This is, in a sense, the "measure-...
1
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1answer
136 views

Do non-measurable sets exist for $\{0,1\}^{\mathbb N}$?

It is well known that a non-Lebesgue measurable, and hence non Borel measurable subset of $[0,1]$ exists. However, if I consider the set $\Omega=\{0,1\}^{\mathbb N}$ in the infinite independent coin ...
0
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1answer
90 views

Borel and Lebesgue measurable sets

How can i prove that the cardinality of Borel sets is less than the cardinality of lebesgue measurable sets?
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0answers
37 views

How many Borel sets are on an infinite Polish Space?

I am learning Descriptive set theory, and my question is how many Borel sets are on an infinite Polish space? I think they have the cardinality of the continuum, but I just do not see how to prove it....
1
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1answer
922 views

Borel sigma algebra - why smallest?

I was wondering why Borel algebra $B(X)$ is defined to be the smallest sigma algebra containing all open subsets of X. If it contains all the subsets, then how can any other sigma algebra have more ...
2
votes
1answer
48 views

Existence of non-Borel subset

I got the following statement: For each set $N \subseteq \mathbb R^n$ with cardinality of the continuum $\#N=\mathfrak c$ there is a subset $M \subseteq N$ with $M \notin \mathcal B(\mathbb R^n)$. ...
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0answers
208 views

How to prove that the cardinality of the set of Borel sets is continuum? [duplicate]

Let $\mathscr{B}$ be the set of Borel sets on $R$, how to prove that $\overline{\overline{\mathscr{B}}}=\aleph$? That is , the cardinality of it is continuum?
2
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2answers
2k views

What's the difference between algebra and $\sigma$-algebra?

The title is quite misleading, I don't have a better one though. It's clear by definition that $\sigma$-algebra is also an algebra. Here is my question, for those algebras which are not $\sigma$-...
3
votes
3answers
108 views

non-Borel subset of uncountable Tychonoff space

Let $X$ be an uncountable Tychonoff space. Must there exist a non-Borel subset of $X$?
1
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1answer
2k views

Cardinality Of Borel Sets

I was trying to show that Borel $\sigma$ algebra is smaller than lebesgue measurable sets. I could come up with a proof for the cardinality of lebesgue measurable sets being $2^c$. Cardinality of ...
1
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0answers
92 views

About the cardinality of the Borelians in the real line. [duplicate]

Is true that the set of the Borelians in $\mathbb{R}$ has the same cardinality of $\mathbb{R}$? I need of the Continuum Hypothesis for to prove this?

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