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?
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### 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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### 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}(\...
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### 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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### 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 ...
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### 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 ...
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### 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 $\sigma$-...
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### 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-...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### non-Borel subset of uncountable Tychonoff space

Let $X$ be an uncountable Tychonoff space. Must there exist a non-Borel subset of $X$?
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### 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 $\tau_U$ ...
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### 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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### 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 ...
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### 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 ...
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### 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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### 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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### 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 $F_E:=\{(x,y)\in X\times X:y\in E_x\}$?...
### 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 ...