My book states that the Lebesgue measure is complete, but does not give a proof. Is the proof difficult?
What I know about the Lebesgue Measure_
All finite half open intervals $(a,b]$
All infinite intervals $(\infty,b]$, and $(a,\infty)$.
I also know that it contains the Borel sigma algebra, that is the sigma algebra generated by all open sets.
Here is how I try to prove it but I get stuck:
I need to prove that if $\mu(A)=0$, and $B \subseteq A$. Then B is also in the sigma algebra.
I start with the set $A$ and assume it is 0, and $B$ is any subset of $A$. Since the measure of $A$ is zero, I know that it can not contain any type of interval, because then the measure would be not 0. But $A$ can consist of points. Here is where I get stuck.
If I can prove that $A$ only can consist of a countable number of points, then it is easy to see that $B$ can also be only a countable number of points. And a each point is in the sigma-algebra, and a countable union is also in the sigma algebra.
But how do I know that $A$ only can consist of a countable number of points?(Or maybe it can be larger, how do I prove it then?)