TITLE: Completeness of Measure spaces
QUESTION [6 upvotes]: 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.
Questions:
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?

REPLY [3 votes]: 3) Take the sample space to be $\Omega=\{1,2,3\}$, the $\sigma$-algebra to be $\mathcal F=\{\emptyset,\Omega,\{1,2\},\{3\}\}$, and let $P$ be the probability measure on $(\Omega,\mathcal F)$ such that $P(\{3\})=1$. Then $P(\{1,2\})=0$ and $\{1\}$ is a non-$\mathcal F$-measurable subset of $\{1,2\}$. The probability space $(\Omega,\mathcal F,P)$ is not complete.