# iterated integral

• Jan 19th 2010, 09:21 PM
PRLM
iterated integral
Let $\lambda,\mu$ be Borel measures on $I=[0,1]$ for x-axis and y-axis. Let $d=\{(x,y): x.
a)show $d$ is measurable on $I\times I$.
b)Evaluate $\int_I \int_I 1_d d\lambda d\mu$, $\int_I \int_I 1_d d\mu d\lambda$, and $\int_{I\times I} 1_d d(\lambda \times \mu)$.

i have no idea how to do this. any help would be appreciated.
• Jan 20th 2010, 04:45 AM
girdav
Let $f\left(x,y\right) =x-y$, for $x,y\in \mathbb R$ We have
$d= f^{-1}\left(\left]-\infty,0\right[\right)\cap I\times I$ and because $f$ is measurable we have the result.
• Jan 23rd 2010, 10:18 PM
PRLM
Quote:

Originally Posted by girdav
Let $f\left(x,y\right) =x-y$, for $x,y\in \mathbb R$ We have
$d= f^{-1}\left(\left]-\infty,0\right[\right)\cap I\times I$ and because $f$ is measurable we have the result.

how do you show that $f$ is a measurable function? and i am still not sure how to do these iterated integrals. please help.
• Jan 24th 2010, 01:26 AM
girdav
It's the sum of two measurable functions.
I think you can use Fubini because you have a $\sigma$-finite space $\left(I,\mu\right)$.