Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Transformation of lebesgue integral

  1. Dec 5, 2008 #1
    define F(x)=x, then uF is the lebesgue stiljes measure
    duF=dx
    Let y=x^2
    we all know that how to transform [tex]\int f(y)dy [/tex] into [tex]\int f(x^2)2xdx [/tex] (***)
    But how exactly would one use the transformation theorem ?
    Ie. T be a measurable transformation from X to Y, u is a measure on X
    [tex]\int_{Y}fduT^{-1}=\int_{X}fTdu[/tex]

    I want to see the transformation theorem in action otherwise it's too abstract for me to understand. My question is, how would one use the transformation theorem to obtain the same result as equation (***)?
     
  2. jcsd
  3. Dec 5, 2008 #2

    morphism

    User Avatar
    Science Advisor
    Homework Helper

    You should ask yourself the following question: what are X, Y and T in this case?
     
  4. Dec 5, 2008 #3
    X is R
    Y is R
    T is sqrt(y)
    T-1 is y^2

    then?
    [tex]\int_{Y}fduT^{-1}=\int_{X}fdu y^2 ? [/tex] doesn't make a lot of sense

    what is the explicit expression of duT^-1 in this example?
     
  5. Dec 5, 2008 #4

    morphism

    User Avatar
    Science Advisor
    Homework Helper

    You need a Radon-Nikodym derivative in there. We have,

    [tex]\int g(T(x)) d\mu(x) = \int g(y) \frac{d(\mu T^{-1})}{d\mu}(y) \: d\mu(y).[/tex]
     
  6. Nov 10, 2009 #5
    I wonder whether someone could write the next step on the solution of this problem. I must confess that I didn't figure out how to use Radon-Nikodym derivative in this case.

    Many thanks
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Transformation of lebesgue integral
  1. Lebesgue integral (Replies: 2)

  2. Lebesgue integral (Replies: 3)

  3. Lebesgue integration (Replies: 3)

  4. Lebesgue Integration (Replies: 3)

  5. Lebesgue integration (Replies: 2)

Loading...