2
votes
0answers
32 views

What is exactly meant by a countable collection in the defintion of a sigma-algebra

This a just a small question about the definition of a sigma-algebra and what is eaxactly meant by countable? Would be grateful for any clarification on this. Most texts define a sigma-algebra ...
0
votes
1answer
84 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
vote
1answer
62 views

Cardinality of cantor set $K$

Is the Cantor function bijective from $[0,1]$ to Cantor set $K$? As $K$ is uncountable I think cardinality of $K$ must be $\mathfrak c$ as $K$ is a subset of $[0,1]$. But I am surprised whether there ...
8
votes
2answers
205 views

Does $\mathcal P ( \mathbb R ) \otimes \mathcal P ( \mathbb R ) = \mathcal P ( \mathbb R \times \mathbb R )$?

Obviously the answer's yes if I replace $\mathbb R$ by a countable set. If $\mathbb R$ is replaced by a set $X$ having a greater cardinality, then the diagonal $\{ (x,x) , \ x\in X\}$ is not in the ...
10
votes
3answers
338 views

Can sets of cardinality $\aleph_1$ have nonzero measure?

$\aleph_1$ is the cardinality of the countable ordinals. It is the least cardinal number greater than $\aleph_0$, and assuming the continuum hypothesis it's equal to $\mathfrak{c}$, the cardinality of ...
1
vote
2answers
87 views

Cardinality of the class of $G_\delta$ subset of $\mathbb{R}$ of Lebesgue measure zero

Let $\mathcal{N}$ be the class of all subsets of $\mathbb{R}$ of Lebesgue measure zero and let $\mathcal{G}_\delta$ be the class of all $G_\delta$ subsets of $\mathbb{R}$. How do I show that ...
1
vote
1answer
584 views

minimal infinite sigma algebra [duplicate]

Does there exist sigma algebra whose cardinality is countably infinite? If yes tell me some examples. If not how to show every infinite sigma algebra is uncountable?
4
votes
4answers
426 views

Cardinality of Vitali sets: countably or uncountably infinite?

I am a bit confused about the cardinality of the Vitali sets. Just a quick background on what I gather about their construction so far: We divide the real interval $[0,1]$ into an uncountable number ...
3
votes
1answer
144 views

Measurable cardinal existence condition

We know that a cardinal $k > \omega$ is measurable if there is a measure function $\mu :2^k\mapsto \{0, 1\}$ that satisfies the following 3 conditions: 1.$<k$ -additive: for every set I of ...
0
votes
1answer
212 views

Does an upper bound on the cardinality of a Vitali set solve the Continuum Hypothesis? [closed]

I set up the background for the question(s) linking the cardinality of the Vitali set to the Continuum Hypothesis. BACKGROUND The Vitali set is proven--based on arguments of Cantor--to be not ...
16
votes
1answer
692 views

Medial Limit of Mokobodzki (case of Banach Limit)

A classical Banach limit is very usefull concept for me, but there is a problem with the integration and even with the measurability, this means for a sequence $(f_n)_{n\in \mathbb{N}}$ of measurable ...