# iterated integral

• Jan 19th 2010, 08:21 PM
PRLM
iterated integral
Let $\displaystyle \lambda,\mu$ be Borel measures on $\displaystyle I=[0,1]$ for x-axis and y-axis. Let $\displaystyle d=\{(x,y): x<y\}\subset I\times I$.
a)show $\displaystyle d$ is measurable on $\displaystyle I\times I$.
b)Evaluate $\displaystyle \int_I \int_I 1_d d\lambda d\mu$, $\displaystyle \int_I \int_I 1_d d\mu d\lambda$, and $\displaystyle \int_{I\times I} 1_d d(\lambda \times \mu)$.

i have no idea how to do this. any help would be appreciated.
• Jan 20th 2010, 03:45 AM
girdav
Let $\displaystyle f\left(x,y\right) =x-y$, for $\displaystyle x,y\in \mathbb R$ We have
$\displaystyle d= f^{-1}\left(\left]-\infty,0\right[\right)\cap I\times I$ and because $\displaystyle f$ is measurable we have the result.
• Jan 23rd 2010, 09:18 PM
PRLM
Quote:

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

how do you show that $\displaystyle f$ is a measurable function? and i am still not sure how to do these iterated integrals. please help.
• Jan 24th 2010, 12:26 AM
girdav
It's the sum of two measurable functions.
I think you can use Fubini because you have a $\displaystyle \sigma$-finite space $\displaystyle \left(I,\mu\right)$.