Let and be two measurable spaces.
Let , where .
For , let . If we now let be a pairwise disjoint sequence in , then the are also pairwise disjoint. <- HERE IS MY QUESTION.
How can this be true? For example, if and , then and are disjoint, because no ordered pair is in both and . But, and ?
Oops, I was wrong first time, and the book is right. The fact that x appears in the notation C(x) implies that x is meant to be fixed. So the sets are the slices of the sets obtained by fixing the first coordinate x and letting the y coordinate vary.
In that example, you should have taken the sets to be and (in other words, the parameter x does not appear in the names of the product sets). If you now choose x = 1.5, for example, then , but the set is empty, because does not contain any points with x-coordinate 1.5.