I am trying to prove the uniqueness of the Riesz representation theory for the real analysis context. The question statement is
Given $\alpha\in BV[a,b]$, show that there is a unique $\beta\in BV[a, b]$ with $\beta(a) = 0$ such that $\beta$ is right-continuous on $(a, b)$ and $\int_a^bfd\alpha = \int_a^bfd\beta$ for every $f\in C[a,b]$.
The integral here are Riemann-Stieltjes inegrals. I have no idea how to approach this problem. Many thanks to any help.