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Math Help - Riemann Stieltjes

  1. #1
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    Riemann Stieltjes

    Let \alpha be an increasing function on [a,b], \quad f,g \in \mathcal{R}[\alpha;[a,b]] and g is non negative on [a,b]. Suppose m= \inf \{f(x): a \leq x \leq b \} and M= \sup \{f(x) : a \leq x \leq b\}. Show that there is a \lambda \in [m,M] such that  \int\limits_{a}^{b} fg \ d\alpha = \lambda \int\limits_{a}^{b} g \ d\alpha
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Chandru1 View Post
    Let \alpha be an increasing function on [a,b], \quad f,g \in \mathcal{R}[\alpha;[a,b]] and g is non negative on [a,b]. Suppose m= \inf \{f(x): a \leq x \leq b \} and M= \sup \{f(x) : a \leq x \leq b\}. Show that there is a \lambda \in [m,M] such that  \int\limits_{a}^{b} fg \ d\alpha = \lambda \int\limits_{a}^{b} g \ d\alpha
    What have you tried? Also, are there stipulations missing?
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  3. #3
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    Attempt

    Hi-

    I thought of applying the Mean Value theorem. But i dont know,...i am not geting it. Please help me out.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Chandru1 View Post
    Hi-

    I thought of applying the Mean Value theorem. But i dont know,...i am not geting it. Please help me out.
    Well...if you know the mean-value theorem for integrals...this is kind of immediate. Do you?
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