A metric space X is called complete if every Cauchy sequence of points in X has a limit that is also in X. It's perfectly clear to me.
A measure space $(X, \chi, \mu)$ is complete if the $\sigma$-algebra contains all subsets of sets of measure zero. That is, $(X, \chi, \mu)$ is complete if $N \in \chi$, $\mu (N) = 0$ and $A \subseteq N$ imply $A \in \chi$. Technically, I could understand the definition, but can't get the logic behind it.
1) Why do we care only about subsets of sets of measure zero to determine completeness?
2) How does the completeness of measure spaces relate to a completeness of metric spaces?
3) Could you suggest a concrete elementary example of a measure space (preferably, with simple sets) that isn't initially complete and then is completed?