$\{(a,b) \in A \times B : f(a)=g(b)\} \in \mathcal A \otimes \mathcal B$ Let $(A,\mathcal A), (B, \mathcal B), (C, \mathcal C)$ be measure spaces. Let $f:A\rightarrow C$ and $g :B\rightarrow C$ be mensurable functions. Let's also suppose that $D=\\{(c,c): c \in C\\} \in \mathcal C \otimes\mathcal C.$ Show that $$E=\\{(a,b) \in A \times B : f(a)=g(b)\\} \in \mathcal A \otimes \mathcal B.$$

Since $D \in \mathcal C \otimes \mathcal C,$ it's easy to see that if we define $h:A\times B\rightarrow C \times C $ by $h(a,b)=(f(a),g(b))$, then $h$ is mensurable and since $h^{-1}(D)=E$ and $D$ is mensurable we get that $E$ is mensurable, that is, $E \in \mathcal A \otimes \mathcal B$.