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**TheProphet** Let $\displaystyle (E,\mathcal{E})$ and $\displaystyle (F,\mathcal{F}) $ be two measurable spaces.

Let $\displaystyle \mathcal{E} \otimes \mathcal{F} = \sigma(\mathcal{E} \times \mathcal{F})$, where $\displaystyle \mathcal{E} \times \mathcal{F} = \{A \subset E \times F : A = \Lambda \times \Gamma, \Lambda \in \mathcal{E}, \Gamma \in \mathcal{F}\} $.

For $\displaystyle C \in \mathcal{E} \otimes \mathcal{F} $, let $\displaystyle C(x) = \{y : (x,y) \in C \} $. If we now let $\displaystyle C_{n} $ be a pairwise disjoint sequence in $\displaystyle \mathcal{E} \otimes \mathcal{F} $, then the $\displaystyle C_{n}(x) $ are also pairwise disjoint. <- HERE IS MY QUESTION.

How can this be true? For example, if $\displaystyle C_{1}(x) = (1,2) \times (5,6) $ and $\displaystyle C_{2}(x) = (3,4) \times (5,6) $, then $\displaystyle C_{1} $ and $\displaystyle C_2 $ are disjoint, because no ordered pair $\displaystyle (a,b)$ is in both $\displaystyle C_1$ and $\displaystyle C_2$. But, $\displaystyle C_{1}(x) = (5,6) $ and $\displaystyle C_2(x) = (5,6) $ ?