Prove that μ is regular on the class of elementary sets E

Question

Let α be a monotonically increasing function defined for all x∈R such that $μ([a,b))=α(b-)-α(a-), μ([a,b])=α(b+)-α(a-), μ((a,b])=α(b+)-α(a+), μ((a,b))=α(b-)-α(a+)$ and let $μ(I_1∪…∪I_n )=μ(I_1 )+⋯+μ(I_n)$ whenever the intervals are pairwise disjoint. Prove that μ is regular on the class of elementary sets E.