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  1. #1
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    Riemann integrable

    Anyone know more about reimann-stieltjes integrable functions? I'm researching them and i need to show if g' is integrable with f on [a,b] continuous, then


    \int_a^b f(x)dg(x) exists and = \int^b_a f(x)g'(x)
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  2. #2
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    Quote Originally Posted by BBrown View Post
    Anyone know more about reimann-stieltjes integrable functions? I'm researching them and i need to show if g' is integrable with f on [a,b] continuous, then


    \int_a^b f(x)dg(x) exists and = \int^b_a f(x)g'(x)

    Look at each summand defining \int_a^b f(x)dg(x) : it is of the form  f(c_i)(g(x_{i+1})-g(x_i)) , where [x_i,\,x_{i+1}] is a subinterval of [a,\,b] and c_i\in [x_i,\,x_{i+1}] .

    Well, now apply the mean value theorem to the function g\;:\;g(x_{i+1})-g(x_i)=g'(d_i)(x_{i+1}-x_i)\,,\,\,d_i\in (x_i,\,x_{i+1}) , substitute this in every summand of

    the sum defining \int_a^b f(x)dg(x) and get the Riemann sum of exactly what you need since we're given g'(x) is Riemann integrable...

    Tonio
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    Quote Originally Posted by tonio View Post
    Look at each summand defining \int_a^b f(x)dg(x) : it is of the form  f(c_i)(g(x_{i+1})-g(x_i)) , where [x_i,\,x_{i+1}] is a subinterval of [a,\,b] and c_i\in [x_i,\,x_{i+1}] .
    Well, now apply the mean value theorem to the function g\;:\;g(x_{i+1})-g(x_i)=g'(d_i)(x_{i+1}-x_i)\,,\,\,d_i\in (x_i,\,x_{i+1}) , substitute this in every summand of
    the sum defining \int_a^b f(x)dg(x) and get the Riemann sum of exactly what you need since we're given g'(x) is Riemann integrable...
    All of the above is true provided we know that \int_a^b f(x)dg(x) exists.
    Do we know it exists?
    Because g' is Riemann integrable implies that it is bounded on [a,b].
    So g has the Lipschitz Condition.
    Thus g is of bounded variation.
    Thus that integral does exist.
    Last edited by Plato; May 7th 2010 at 02:45 PM.
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    Quote Originally Posted by Plato View Post
    All of the above is true provided we know that \int_a^b f(x)dg(x) exists.
    Do we know it exists?
    We need to more about f and/or g.

    Well, f is cont. and thus Riemann integ., and we're given g' is (Riemann) int., so fg' is Riemann int., and my first post proves the Riemann-Stieltjes int. on the left equals the Riemann int. on the right, so from the existence of the RHS and the equality between both sides it follows the existence of the LHS.

    Tonio
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    Thank you so much... I should have known: when in doubt, apply the mean value thm. Thanks for the guidance I truly appreciate it.
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    Quote Originally Posted by tonio View Post
    Well, f is cont. and thus Riemann integ., and we're given g' is (Riemann) int., so fg' is Riemann int., and my first post proves the Riemann-Stieltjes int. on the left equals the Riemann int. on the right, so from the existence of the RHS and the equality between both sides it follows the existence of the LHS.
    Frankly I do not find that argument persuasive.
    I am not sure that one approximating sum can prove the existence of the integral.
    Here is my point: A necessary and sufficient condition that \int_a^b {fdg} exist for every function f continuous on [a,b] is that g be of bounded variation on [a,b]
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    Quote Originally Posted by Plato View Post
    Frankly I do not find that argument persuasive.
    I am not sure that one approximating sum can prove the existence of the integral.
    Here is my point: A necessary and sufficient condition that \int_a^b {fdg} exist for every function f continuous on [a,b] is that g be of bounded variation on [a,b]

    I think my argument is sound, and there's no approximation in it: it shows that the sum whose limit defines the Riemann-Stieltjes integral on the LHS equals the sum whose limit limit defines the Riemann integral on the RHS, and when we pass to the limit in both sides (when the number of partition points goes to infinity and the partition's length goes to zero), this last sum's limit exists since fg' is a Riemann integrable function, as already said, from which also follows the existence of the LHS's linit.

    The bounded variation condition on g is fine to know beforehand that the LHS integral exists (together with f being continuous or simply Riemann integrable), but I think we can also deduce its existence by the above argument.


    Tonio
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  8. #8
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    Either of you looked at AidenMatthew's post about integrability? He helped me out before and I'd like to help him but I think I'm making the problem easier than it is. Ideas?
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