# Linked Questions

22 questions linked to/from Cardinality of Borel sigma algebra
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 $F_E:=\{(x,y)\in X\times X:y\in E_x\}$?...
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 ...
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 ...
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 ...
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-...
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 ...
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?
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....
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 ...
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 $M \notin \mathcal B(\mathbb R^n)$. ...
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?
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 $\sigma$-...
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 $X$?
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 ...
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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