I'm dealing with this case:
Let $(X,\mathcal{X},\mu), (Y,\mathcal{Y},\nu)$ measurable spaces, with $X=Y=[0,1],\mathcal{X}=\mathcal{Y}=$ Borel in $[0,1].$ Let $\mu$ the Lebesgue Measure and $\nu$ the counting measure.
Now, let $D=\{(x,y)\in X\times Y:x=y\}.$ I'm trying to calculate the measure of $D$ with the product measure $\pi.$
I know that $\pi(A\times B)=\mu(A)\times\nu(B),$ but I can't write $D$ as a product of two rectangles in $Z=X\times Y$. So, I'm trying to build some sequences $(A_{n})\in\mathcal{X},(B_{n})\in\mathcal{Y}$ such that $D=\lim(A_{n}\times B_{n})$, but I didn't get it. What can I do?
EDIT: Remember that $(Y,\mathcal{Y},\nu)$ is not a $\sigma-$finite space, so I cannot use Fubini or Tonelli here. Not even the following Lemma:
