Linked Questions

39
votes
3answers
10k views

The $\sigma$-algebra of subsets of $X$ generated by a set $\mathcal{A}$ is the smallest sigma algebra including $\mathcal{A}$

I am struggling to understand why it should be that the $\sigma$-algebra of subsets of $X$ generated by $\mathcal{A}$ should be the smallest $\sigma$-algebra of subsets of $X$ including $\mathcal{A}$. ...
50
votes
2answers
10k views

Cardinality of Borel sigma algebra

It seems it's well known that if a sigma algebra is generated by countably many sets, then the cardinality of it is either finite or $c$ (the cardinality of continuum). But it seems hard to prove it, ...
44
votes
2answers
12k views

Lebesgue measurable but not Borel measurable

I'm trying to find a set which is Lebesgue measurable but not Borel measurable. So I was thinking of taking a Lebesgue set of measure zero and intersecting it with something so that the result is not ...
6
votes
5answers
9k views

Achilles and the tortoise paradox?

Let's say we decide to race on a track $1000$ km long. You are a $100$ times faster than me, meaning if we both start at the beginning, you obviously win. To make things more fair you give me a head ...
11
votes
2answers
3k views

between Borel $\sigma$ algebra and Lebesgue $\sigma$ algebra, are there any other $\sigma$ algebra?

Is there any $\sigma$-algebra that is strictly between the Borel $\sigma$-algebra and the Lebesgue $\sigma$-algebra? How about not in between the two, but in general, are there any other $\sigma$ ...
22
votes
1answer
3k views

Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly

While reading David Williams's "Probability with Martingales", the following statement caught my fancy: Every subset of $\mathbb{R}$ which we meet in everyday use is an element of Borel $\sigma$-...
1
vote
2answers
2k views

Measurable Maps and Continuous Functions

I'm having a bit of trouble understanding the connection between measurable maps and continuous functions. Namely, I know the following: Assume $f:(\Omega_{1}, \mathcal{B}_{1}) \rightarrow (\Omega_{2}...
1
vote
1answer
134 views

How many subsets of $\mathbb R$ are not Lebesgue-measurable?

Moving from the Borel sigma algebra $\mathfrak B(\mathbb R)$ to the Lebesgue sigma algebra $\mathfrak L(\mathbb R)$ increases the number of mearurable sets substantially: $\mathfrak B(\mathbb R)$ has ...
0
votes
0answers
176 views

Measurable non-analytic set

I know the construction by Lusin of a measurable set that is non-Borel. But that set turns out to be analytic. Are there some examples of non-analytic sets that are measurable? Maybe the set that one ...
0
votes
1answer
37 views

Is there anything else than a $\sigma$-algebra or is $\sigma$-algebra the only meaningful algebra on sets?

Is there anything else than a $\sigma$-algebra or is $\sigma$-algebra the only meaningful algebra on sets? It seems that the $\sigma$-algebra has been invented in order to serve some particular ...