Sigma algebras generated by two different generators. I have just started my study of advanced probability theory with the book by Klenke. The author mentioned two generators in an example:


*

*$\mathcal{E}=\{E_n:n\in\mathbb{Z}\}$ where $E_n=[-n,n]\cap\mathbb{Z}$.

*$\mathcal{F}=\{F_n:n\in\mathbb{Z}\}$ where $F_n=[-\frac{n}{2},\frac{n+1}{2}]\cap\mathbb{Z}$.


According to the author, the $\sigma(\mathcal{E})\neq2^{\mathbb{Z}}$ while $\sigma(\mathcal{F})=2^{\mathbb{Z}}$.
Since I never enrolled courses involved with measure theory, understanding the differences between the two generators and hence the sigma algebra is somewhat difficult for me. My questions are:
1.What sigma algebra is generated by the first generator?
2.What differences between these two generators contribute to the different sigma algebras?
Thank you for your help!
 A: In the second case $\{-n\} =F_{m+1}-F_m$ where $m=2n-1$. I will you verify that $\{n\}$ is also in the  sigma algebra for $n \geq 0$.  Hence every singleton set is in $\sigma (\mathcal F)$ which makes $\sigma (\mathcal F)$  the power set of $\mathbb Z$. 
$\sigma (\mathcal E)$ is the class of al symmetric subsets of $\mathbb Z$.
A: $A\subseteq\mathbb Z$ is by definition a symmetric set if for every $n\in A$ we also have $-n\in A$.
Let $\mathcal S:=\{A\in2^{\mathbb Z}\mid A\text{ is symmetric}\}$.
It is not difficult to prove that $\mathcal S$ is a $\sigma$-algebra and this with $\mathcal E\subseteq\mathcal S$.
This allows the conclusion that $\sigma(\mathcal E)\subseteq\mathcal S$.
Moreover it is evident that $\mathcal S\neq2^{\mathbb Z}$ and consequently $\sigma(\mathcal E)\neq2^{\mathbb Z}$.

Let us have a closer look at the $F_n$ and check for a pattern to make things more easy:


*

*$F_0=\{0\}$

*$F_1=\{0,1\}$

*$F_2=\{-1,0,1\}$

*$F_3=\{-1,0,1,2\}$

*$F_4=\{-2,-1,0,1,2\}$

*$F_5=\{-2,-1,0,1,2,3\}$
Now observe that the sets $F_0$ and $F_n-F_{n-1}$ for $n=1,2,\dots$ are all singletons and that moreover every singleton set can be written in such a way.
That means that every singleton is an element of $\sigma(\mathcal F)$.
Every subset of $\mathbb Z$ can be written as a countable union of singletons so will be an element of $\sigma(\mathcal F)$.
Proved is now that $\sigma(\mathcal F)=2^{\mathbb Z}$.
