Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Give an example of sigma algebra in $\mathcal P(\Bbb N)$ whose order is finite, and also whose order is infinite, and whose order is $10$?

(the order means the number of elements)

share|cite|improve this question

closed as off-topic by 6005, saz, Solid Snake, Kevin Dong, drhab Aug 31 at 10:16

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – 6005, saz, Solid Snake, Kevin Dong, drhab
If this question can be reworded to fit the rules in the help center, please edit the question.

What is the order of a $\sigma$-algebra? If it is the number of elements, there is no $\sigma$-algebra with exactly $10$ elements. – Michael Greinecker Oct 20 '12 at 7:19
why is that?and yes the order is the number of element – Dania Nabeel Hadieh Oct 20 '12 at 8:02
Well, well. What a lazy question. Nonetheless plus one because I learned something from reading Davide's answer. – Rudy the Reindeer Oct 20 '12 at 9:36
Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. If this is homework, please add the homework tag; people will still help, so don't worry. Also, many find the use of imperative ("Prove", "Solve", etc.) to be rude when asking for help; please consider rewriting your post. – Julian Kuelshammer Oct 21 '12 at 19:38

1 Answer 1

A example of $\sigma$-algebra contained in $\mathcal P(\Bbb N)$ which is finite is $\{\emptyset,\Bbb N\}$; an example of infinite one is $\mathcal P(\Bbb N)$ itself.

Now let $X$ a set and assume that $\mathcal B$ is a finite $\sigma$-algebra on $X$. For each $x\in X$, define $S_x:=\bigcap_{B\in\mathcal B, x\in B}B$, and define $x\sim y$ if $x\in S_y$. Then $\sim$ is an equivalence relation, which gives a finite partition of $X$ as $S_{x_1},\dots,S_{x_n}$ (each $S_{x_i}$ is measurable and the $\sigma$-algebra is assumed finite). So $\mathcal B$ has $2^n$ elements.

If we take an infinite set $S$, we can, for each $n\geq 1$, construct a $\sigma$-algebra having exactly $2^n$ elements (taking a partition of $S$ in $n$ elements).

share|cite|improve this answer

Not the answer you're looking for? Browse other questions tagged or ask your own question.