Using Fubini's Theorem

**akolman** · October 17th 2011, 09:08 PM

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,

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