Thread: Ask a easy question

This question comes from page 25, line 4, of Halmos' "Measure Theory", as the image below shows. This line says that is a -ring which can be verified easily, but I just can not verify this easy statement because I can not make up the difference into a form of . Could you please help me with this problem? Thanks!

NOTE: Here means the -ring generated by .

To show that is a -ring, it is necessary to prove that it is closed under at most countable union and difference. The union part is really very easy and we prove the difference part. Suppose are two arbitrary elements of where and , we need to show that their difference also assumes the form of and therefore belongs to . By taking at most countable union of sets in , the Theorem B in page 22 of this book can be extended to -ring case, that is, If is any class of sets, then every set in can be covered by at most countable union of sets in . So where , it means , so is disjoint from , and both is disjoint from both and by the same argument. It can be proved that if both sets and are disjoint from both sets and , we have , so . Because is in and is in by definition of -ring, the result above is of the desired form and the easy statement is verified not very easily.