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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)

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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
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Well, well. What a lazy question. Nonetheless plus one because I learned something from reading Davide's answer. –  Matt N. Oct 20 '12 at 9:36
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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).

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