Question: If f(x) is Riemann-Stieltjes integrable function with respect to g(x) on [a,b]and if |f(x)dg(x) =0 for every function f(x) which is monotonic on [a,b]

If f(x) is Riemann-Stieltjes integrable function with respect to g(x) on [a,b]and

If f(x) is Riemann-Stieltjes integrable function with respect to g(x) on [a,b]and if |f(x)dg(x) =0 for every function f(x) which is monotonic on [a,b] . Prove that g(x) must be constant on [a,b].

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