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Math Help - questions about lebesgue measure

  1. #1
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    questions about lebesgue measure

    (X,F,\mu)=([-1,1],\Lambda,\lambda) , where \Lambda is a sigma algebra of Lebesgue measurable sets and \lambda is Lebesgue measure. Define v(A)= \int_A x d\lambda for all A \in \Lambda.

    a) prove v is a signed measure.
    b) prove v << \lambda
    c)Find the Hahn-Jordan decomposition of v and its total variation.

    i dont even know how to get start. please help me. and im not sure what << means on b).
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  2. #2
    Senior Member Shanks's Avatar
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    1) varify that v satisfy the definition of sigh measure.
    2) the v << u notation mean v is absolutely continous with respect to u, that is, if A is a measureable set with u(A)=0, then v(A)=0.
    3)make sure that you understand the Hahn-Jordan decomposition and total variation of a measure in your textbook.
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  3. #3
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    Quote Originally Posted by Shanks View Post
    1) varify that v satisfy the definition of sigh measure.
    2) the v << u notation mean v is absolutely continous with respect to u, that is, if A is a measureable set with u(A)=0, then v(A)=0.
    3)make sure that you understand the Hahn-Jordan decomposition and total variation of a measure in your textbook.
    on 1), the definition says that v assumes at most one of \infty, -\infty. but how do i compute this integral? \int_A x d\lambda \neq \int_A x dx, correct? how do i compute the smallest value and the biggest value it assumes? could you help me on this?
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  4. #4
    Senior Member Shanks's Avatar
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    the equality holds since they define the same integral in the sense of lebesgue integration.
    for the second problem,let f(x)=x, what is the positive part of f, and the negative part? the biggesr value it assumes is the integral of the positive part of f, similarly for the smallest value.
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    Quote Originally Posted by PRLM View Post
    on 1), the definition says that v assumes at most one of \infty, -\infty. but how do i compute this integral? \int_A x d\lambda \neq \int_A x dx, correct? how do i compute the smallest value and the biggest value it assumes? could you help me on this?
    i am really not sure if this is correct so i would like someone to confirm what i did is correct, please.

    1). to show that v is a signed measure, i verified that v checked the definition.
    a)v assumes at most one of \infty ,-\infty.
    vA=\int_A x d\lambda = \int_A x dx since x is bounded on a finite interval and it is reimann integrable. and since the integral is all finite on any set A \subset [-1,1], v does not assume any of \infty,-\infty.
    b) v(\emptyset)=0 since the integral over a set of measure zero is zero.
    c) by the equation \int \Sigma f =\Sigma \int f, it is easy to verify.
    Therefore it is a signed measure.

    2).To show |v|<<\mu, show that for each subset A such that \mu A=0, |v|A=0. again since the integral over a set of measure zero is zero, it is obvious.

    i am not sure what \frac{dv}{d\mu} is. but i assume that it is Randon Nikodym derivative. Since v is a signed measure, by restricting v to a positive set, we get a measure. So v is restricted to [0,1]. So \frac{dv}{d\mu} = x \in [0,1].

    3)Hahn-Jordan decomposition is v^+(E)=v(E \cap [0,1]),v^-(E)=-v(E \cup [-1,0)). |v|(X)=\int_0^1 xdx-\int_{-1}^0 xdx = 1.

    please someone correct me if im wrong.
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  6. #6
    Senior Member Shanks's Avatar
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    the Radon Nikodym derivative is f(x)=x, x belongs to [-1,1].
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    Quote Originally Posted by Shanks View Post
    the Radon Nikodym derivative is f(x)=x, x belongs to [-1,1].
    i thought that Radon N derivative is a nonnegative function.
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  8. #8
    Senior Member Shanks's Avatar
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    No, it is possible that the Radon Nikodym derivative assumes negative value.
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  9. #9
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    Quote Originally Posted by Shanks View Post
    No, it is possible that the assumes negative value.
    would you explain how u get Radon Nikodym derivative ?
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  10. #10
    Senior Member Shanks's Avatar
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    One way to do this is by the definition of derivative of a measure with respect to another measure.(it is complicated )
    another easier way is using the Radon Nikodym theorem. it asserts the existence and uniqueness of Radon Nikodym derivative of a measure which is absolutely continous with respect to another measure.
    for more information you can see Rudin's book Real and Complex Analysis.
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