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Math Help - Riemann Stieltjes Integral

  1. #1
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    Riemann Stieltjes Integral

    Let f(x) be continuous on [a,b]. Let \alpha (x) be increasing on [a,b].
    Suppose \int f(x)g(x)d\alpha (x) \equiv 0 for every g(x) such that f(x)g(x) \in R(\alpha).
    Prove f(x) \equiv 0 on [a,b]
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  2. #2
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    Take g(x)=f(x) on [a,b]. Then you just need to show that if w(x)\ge{0}, continuous, and \int_a^bw(x)dx=0 then w(x)\equiv{0} on the interval.

    Hint: Try contradiction, and use the fact that w is continuous.
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