Give an example of sigma algebra in $\mathcal P(\Bbb N)$ 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)

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).
