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hw5 - 5(a Let S be a collection of subsets of[0 1 that is...

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Math 318 HW #5 Due 5:00 PM Thursday, March 3 Reading: § 10–12. Problems: 1. Show that the sets E x defined as part of showing that m * is not countably additive (cf. Example 7.7) form equivalence classes. Use this to conclude that for any x, y [0 , 1], either E x = E y or E x E y = . 2. Exercise 9.8. 3. Exercise 9.27. 4. Let S be the set of all intersections of Q with arbitrary closed, open, and half-open subintervals of [0 , 1], including degenerate closed subintervals consisting of a single point. Define μ : S → R by μ ( A cd ) = d - c, where A cd is the intersection of Q with any of the intervals ( c, d ), [ c, d ], [ c, d ), or ( c, d ]. Show that μ is finitely additive but not countably additive and, hence, not a measure.
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Unformatted text preview: 5. (a) Let S be a collection of subsets of [0 , 1] that is closed under countable unions. Suppose μ : S → R is a set function satisfying the first three conditions for being a measure. In addition, suppose μ is i. finitely additive (meaning μ ( A ) = ∑ n i =1 μ ( A i ) for A = S n i =1 A i with A i ∈ S and A i ∩ A j = ∅ whenever i 6 = j ); and ii. countably sub-additive (meaning μ ( A ) ≤ ∑ ∞ i =1 μ ( A i ) for A = S ∞ i =1 A i with A i ∈ S and A i ∩ A j = ∅ whenever i 6 = j ). Prove that μ is a measure on S . (b) Explain why Exercise 9.29 is an easy corollary of part (a). 1...
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