Let be a function, and let be a -algebra over . Similar to one of the exercises, we prove  is a -algebra over .

If there is (are) mistake(s) in the proof, select all corresponding answers. Select "The proof is correct.", if and only if 1) there are no errors in the proof, and 2) the proof is complete.  

Proof:

(A) In order to prove is a -algebra, we prove all three defining properties.

(B) First, we show .

   (B 1) For this, we note that , as is a -algebra over .

   (B 2) Then, as is a function , it follows and therefore .

(C) Then, we show that for any we have , where we take the complement in .

   (C 1) If , then - by definition of - there exists .

   (C 2) As is a -algebra, from it follows , where the complement has to be taken in .

   (C 3) By the definition of we observe that .

   (C 4) Therefore, as we get .

(D) Finally, we have to show that for any sequence of sets the union .

   (D 1) Take such a sequence , then there must be a corresponding sequence with .

   (D 2) By being a -algebra, it follows that the union of the is contained in , i.e. there is .

   (D 3) Therefore, also .

This concludes the proof.