Finite measure divided into small measures 
Show that if $E$ has finite measure and $\epsilon>0$, then $E$ is the disjoint union of a finite number of measurable sets, each of which has measure at most $\epsilon$.

First, I'm quite confused with the problem statement. If it's true, it means any set $E$ with finite measure is the disjoint union of a finite number of measurable sets, and hence must be measurable. But any set with finite measure might not be measurable, or is it?

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*Prove it when $E$ is a $G_\delta$, that is, a countable intersection of open sets. 

*In the general case, prove that there is a $G_\delta$, say $F$, such that $\lambda(F)=\lambda^*(E)$ and $E\subset F$.
