What is the generated σ -Algebra?

Question

Suppose that $\Omega=\mathbb{Z}$ and let $\mathscr{T}$ be the $\sigma$-algebra generated by $S_n=\{n,n+1,n+2\}$ with $n\in\mathbb{Z}$. What are the elements generated by $\mathscr{T}$?

I noticed that $S_{n-2}\cap S_n=\{n\}$ but I was stuck here.

Answer 1

The $\sigma$-algebra is just $\mathcal{P}(\Omega)$, that is, the power set of $\Omega$ (the set containing all subsets of $\Omega$). Take some $n\in\Bbb Z$. Then $S_n=\{n,n+1,n+2\}$ and $S_{n-2}=\{n-2,n-1,n\}$. Since the $\sigma$-algebra we are considering is generated by this type of sets, both sets should be in that $\sigma$-algebra, and so should be their intersection. So for every $n\in\Bbb Z$ we have that $S_n\cap S_{n-2}=\{n\}$ is in the $\sigma$-algebra. Then, for every subset $A\subset Z$ we have that $A$ is in the $\sigma$-algebra too, since $A=\cup_{n\in A}\{n\}$, which is a countable union of sets of the $\sigma$-algebra (it's countable because $A\subset\Bbb Z$ and $\Bbb Z$ is countable).