Riemman-Stieltjes Integral: Proving Supremum Property

[takeuchi]

Let α>0, J:=[-a,a] and f:J→ℝ a bounded function.
Let α an increases monotonically on J and P* the set of all partitions P of J containing 0 and such that are symmetric,i.e, x in P iff -x in P. Prove that
∫fdα= sup L(P,f,α) with P in P*

 

[boombaby]

It is sufficient to show that for a given e>0, there exist a P* such that ∫fdα-L(P*,f,α)<e.
Find a P first and let P* be a refinement of P, which is also symmetric and containing 0. Then P* is what you want.