Then it said without proof that μ is finitely additive, but not σ-additive.
As I did not get why I tried to prove it by myself and I tried to show that is a semi-ring, I guess that is important before I start with the other proof.
We have and furthermore and the union of two closed intervals is either an interval or the disjoint union of two intervals. Then is also an interval or the disjoint union of two intervals.
Now the proof. I do not quite understand how it cannot be -additive. Does it have something in common with Cantor sets? I don´t know how to start the proof here. Any help or explanation (maybe an idea for the beginning of a proof) is appreciated. If there is a proof...