Let \(\mu(A):=\# A\) be the counting measure and \(\lambda\) be Lebesgue measure on the measurable space \(([0,1], \mathscr{B}[0,1])\). Denote by \(\Delta:=\left\{(x, y) \in[0,1]^{2}: x=yight\}\) the diagonal in \([0,1]^{2}\). Check that

\[\int_{[0,1]} \int_{[0,1]} \mathbb{1}_{\Delta}(x, y) \lambda(d x) \mu(d y) eq \int_{[0,1]} \int_{[0,1]} \mathbb{1}_{\Delta}(x, y) \mu(d y) \lambda(d x)\]

Does this contradict Tonelli's theorem?