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Math Help - Using Fubini's Theorem

  1. #1
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    Acolman, Mexico
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    Using Fubini's Theorem

    Hello, I need help to write formally the answer to the following problem

    Define a measure on \mathcal{R}^2 (the Borel subsets of \mathbb{R}^2) by

    \mu((a,b]\times(c,d]) = I(a < 0 \leq b)[\max\{\min(d,1)-max(c,0),0\}]

    a. let f(x,y)=(1+x)^2. Find \int f d\mu

    I did a graph, and I think this should be 1

    b. let  g(x,y)=y^2. Find \int g d\mu

    Same deal as above, I did a graph and I think it has to be 1/3


    If I split

    \mu((a,b]\times(c,d]) = I(a < 0 \leq b)[\max\{\min(d,1)-max(c,0),0\}]

    into
    \mu_1((a,b])=I(a < 0 \leq b)
    and
    \mu_2((c,d]) = [\max\{\min(d,1)-max(c,0),0\}]

    Both \mu_1 and \mu_2 are \sigma-finite measures, also is \mu_2 the Lebesgue measure restricted to (0,1]?

    Am I on the right track?

    Thanks in advance,
    Last edited by akolman; October 18th 2011 at 06:03 PM.
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