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Thread: 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 $\displaystyle \mathcal{R}^2$ (the Borel subsets of $\displaystyle \mathbb{R}^2$) by

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

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

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

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

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


    If I split

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

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

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

    Am I on the right track?

    Thanks in advance,
    Last edited by akolman; Oct 18th 2011 at 05:03 PM.
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