You missed some cases, for example when both and are in (that's what the example shows), and when neither nor is in . You have to consider the four cases.
I need help with understand a theorem.
Theorem Let , and be any measurable space. An indicator on is measurable if and only if .
Proof: If is measurable then is in . I guess we can see this by noticing that , which is in and thus also by definition.
Now, conversely assume . Then if .
Clearly because is a -algebra. But shouldn't it be if ? By taking for example you get a lot of points where .