The above algebra of sets is used to define a pre-measure. The pre-measure was said to be well-defined and finitely additive from the simple fact:
Each element in $ A$ can be uniquely written as a disjoint union of maximal left-open/right-closed intervals in $\mathbb R$.
I am trying to prove the simple fact but I cant seem to be able to show it.
A possible method: for all $x$ in E which is an element of A I tried to consider the union of all left-open\right-closed intervals in E that contained $x$. However, I don't know how to show the the union is still a half-open interval.(This is similar to showing that every open set is a countable union of disjoint intervals.)
Any ideas or suggestions as to how to prove the simple fact?