# Linked Questions

22 questions linked to/from Cardinality of Borel sigma algebra
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### 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?
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?
4answers
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### Is there a $\sigma$-algebra on $\mathbb{R}$ strictly between the Borel and Lebesgue algebras?

So, after proving that $\mathfrak{B}(\mathbb{R})\subset \mathfrak{L}(\mathbb{R})$, I asked myself, and now asking you, is there a set $\mathfrak{S}(\mathbb{R})$, which satisfies: \mathfrak{B}(\...
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 ...
2answers
10k views

### set in $\mathbb{R}$ which is not a Borel-set [duplicate]

Possible Duplicate: Lebesgue measurable but not Borel measurable Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly if i start from the topology of $\mathbb{R}$, i.e. all open ...
1answer
6k views

### Positive outer measure set and nonmeasurable subset

I'm attending a course of Measure and Integration and have some homework to do. We don't have a specific book to follow, neither for exercise. I'm asked to proof that every set $A \in R$ with ...
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 $\sigma$-...
1answer
477 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
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 ...
2answers
227 views

### Why are there exactly 2$^\omega$ perfect subsets of the real numbers?

How can you proof that there are $2^{\omega}$ perfect subsets of the real numbers?
2answers
850 views

### A set in a $\sigma$-algebra that can't be “reached” with countable set-theoretical operations

Can someone please give me an example of a set that lies in a $\sigma$-algebra generated by some set other then the $\sigma$-algebra itself, such that this (the first) set can't be obtained by ...
1answer
925 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 ...
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 $X$?
2answers
180 views

### Uniform versus product topologies on $[0,1]^\mathbb{N}$, and their Borel $\sigma$-algebras

Let $\tau_U$ and $\tau_P$ be the uniform (i.e. sup-metric) and product topologies on $[0,1]^\mathbb{N}$, respectively. Clearly, these topologies are not the same ($\tau_P$ is separable and $\tau_U$ ...
2answers
255 views

### Any example for a function having domain and range as subset of real line that is NOT Borel function?

Suppose there is a function $f:A\to B$ where $A,\,B\subseteq\mathbb{R}$, is there any example for this function being NOT Borel function? Well the question came up to be when I was reading the ...

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