(a) (I) Since is -algebra, we have , and therefore, .
(II) Suppose that and consider in . We have and , and since -algebra is closed under complement, we get , so that
Thus, is closed under complement.
(III) For any countable collection of subsets in , there is a countable collection of subsets in such that for all . Since is closed under countable union, we get
Therefore,
showing that is closed under countable union.
By (I), (II), and (III), we conclude that is -algebra of in .
b) Since is the generator of -algebra , we have the following two equivalent conditions:
(I) , and
(II) if and is -algebra on then .
Now, to prove that is generator of , we need to prove the following:
(I) , and
(II) if and is -algebra on then .
Proof of (I): If then for some . Since , we have , and hence, . Thus, .
(II) Suppose is -algebra on and . Let be the -algebra on generated by . Then , and hence, . Thus, if then so that . This proves .
Hence, is generator of .
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