Suppose that ( (1, (2, , and let A be the discrete Iuml field On A, define the set function Icirc frac14 by Then show that (i) Icirc frac14 is nonnegative, Icirc frac14 ( Atilde 0 and is finitely additive, but it is not Iuml additive (ii) ( limn( An for a sequence An , with An ( An 1, n ( 1, and Icirc frac14 (An) 0 (and therefore Icirc frac14 ) (Acn) (), n ( 1 0 if A is finite, 11 (A) if A is infinite X

Question: Suppose that ( = {(1, (2,...}, and let A be the discrete Ï-field. On A, define the set function Î¼ by Then show that (i)

Suppose that ( = {(1, (2,...}, and let A be the discrete Ïƒ-field. On A, define the set function Î¼ by

Then show that
(i) Î¼ is nonnegative, Î¼(Ã˜ = 0 and is finitely additive, but it is not Ïƒ-additive.
(ii) ( = limn†’( An for a sequence {An}, with An ( An+1, n ( 1, and Î¼ (An) = 0 (and therefore Î¼) (Acn) = (), n ( 1.

0 if A is finite, 11 (A) if A is infinite. X

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