Let (mu(A):=# A) be the counting measure and (lambda) be Lebesgue measure on the measurable space (([0,1],
Question:
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?
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