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Thread: Riemann Stieltjes Integral

  1. #1
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    Riemann Stieltjes Integral

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

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