I need help with understand a theorem.

TheoremLet , 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 .