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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- Spring '08
- Sets, Empty set, Rational number, Countable set, form equivalence classes