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**GTO** Let $\displaystyle (X,M,\mu)$ and $\displaystyle (Y,N,\nu)$ be measure spaces. Let $\displaystyle X=Y=[0,1]$ with $\displaystyle M=N$ the Borel $\displaystyle \sigma$ algebra. Let $\displaystyle \mu$ be Lebesgue measure and $\displaystyle \nu$ be the counting measure. Define the diagonal in $\displaystyle X$x$\displaystyle Y$ as $\displaystyle D=\{(x,x) : x \in [0,1]\}$, and let $\displaystyle 1_D$ be the indicator function.

Compute the following

$\displaystyle \int_{X \times Y} 1_D d(\mu \times \nu)$, $\displaystyle \int_X \int_Y 1_D d \nu d\mu$, and $\displaystyle \int_Y \int_X 1_D d \mu d\nu$

this is really hard. please help me