Let \((X,\mathcal {M},\mu )\) and \((Y,\mathcal {N},\nu )\) be two measure spaces and \(A\) be measurable with respect to \(\mathcal {M}\times \mathcal {N}\), the \(\sigma \)-algebra generated by measurable rectangles. Show that if \(A_1\subset A_2\subset \cdots \) satisfy that \[ (\mu \otimes \nu )(A_i)=\int _X \nu (\{y:(x,y)\in A_i\})\,d\mu (x) \]
for all \(i\) then
\[ (\mu \otimes \nu )\left (\bigcup _{i=1}^{\infty }A_i\right ) =\int _X \nu \left (\{y:(x,y)\in \bigcup _{i=1}^{\infty }A_i\}\right )\,d\mu (x). \]