Let us consider subsets of the real line. An $F_{\sigma}$ set is a set which can be written as a countable union of closed sets. Suppose $E$ is an $F_{\sigma}$ set. Is it true that there exists a countable collection of pairwise disjoint sets $\left\{E_{k}\right\}_{k}$, such that \begin{equation*} E = \cup_{k} E_{k}? \end{equation*} Obviously the usual measure theory construction of replacing $E_{n}$ by $\tilde{E}_{n} = E_{n} \setminus \cup_{k=1}^{n-1}E_{k}$ fails to preserve closedness.
I am not sure if it plays a role, but I would also be interested in the same question when all $E_{k}$s are assumed to have measure $0$.
EDIT:
So by the comment it does not hold for open sets. What can be said if $E$ has measure $0$?
