About a Possible Decomposition of a Set $E$ of Infinite Measure Can we decompose $m(E)=\infty$ into $E=\bigcup_{n\in \mathbb{Z}}E_n$ where $E_n=E\bigcap[n-1,n]$ where $n \in \mathbb{Z}$? Then we have $\{E_n\}_{n\in \mathbb{Z}}$, a  countable collection of measurable subsets of $E$ and of finite measure since $E_n \subseteq [n-1,n]$ so $m(E_n)\leq 1< \infty$. Also, each pair of $E_n$'s are not necessarily disjoint.
Now, if there is a closed set $F_n \subseteq E_n$ for each $n\in\mathbb{Z}$, this means that the collection $\{F_n\}_{n\in \mathbb{Z}}$ do not have pairwise disjoint sets in it right?
My question might be lacking in motivation, but I was hoping that if my constructions above are true then it might be a remedy in my proof (invalid) on the extension of Lusin's theorem to the case where $m(E)=\infty$ which appears in this post: Lusin's Theorem when $m(E)=+\infty$. Proof Verification.

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*First question: Yes.  From $E_n \triangleq E \cap [n-1,n]$, it's trivial that $E=\bigcup_{n\in \Bbb Z} E_n$.

*Second question: No.  Consider $E = \Bbb R$, then $m(E) = \infty$.  $$E_n \triangleq E \cap [n-1,n] = [n-1,n],$$ so "each pair of $E_n$'s is not necessarily disjoint" (as $E_n \cap E_{n+1} = \{n\}$). $$F_n \triangleq E \bigcap \left [n-\dfrac34,n-\dfrac14 \right] = \left [n-\dfrac34,n-\dfrac14 \right] \subseteq E_n \quad \forall\,n \in \Bbb N,$$  but the collection of closed subsets $\{F_n\}_{n\in \Bbb Z}$ is pairwise disjoint: $$F_i \cap F_j = \varnothing \text{ whenever } i \ne j.$$

