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Unformatted text preview: Math 515 Professor Lieberman September 7, 2004 HOMEWORK # 2 SOLUTIONS Chapter 3 4. To show that n is countably additive, let h E j i be a sequence of disjoint sets. We consider three possibilities. (A complete proof requires a lot more set theory than I want to deal with in this course.) First, only finitely many sets are nonempty and the nonempty sets are all finite. Then the number of elements in the union of these sets is equal to the sum of the number of elements in each set because the sets are disjoint. Second, there is an infinite set in our list. Then the union is infinite, and nE j = . Finally, if there are infinitely many nonempty sets, then the union has infinitely many elements and nE j = , again. If E is a set and y R , then the function f : E E + y defined by f ( x ) = x + y is one-to-one and onto, so E and E + y have the same number of elements, so n is translation invariant....
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