This proves that σ c contains all open sets being a

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). This proves that σ ( C ) contains all open sets, being a σ -algebra containing all open sets, it must contain σ ( O ) = B ( X ). Similarly one sees that B ( X ) = σ ( O ) σ ( O ). 4. Let C be the family of all singleton subsets of R . That is, A ∈ C if and only if A is a set consisting of a single element. Describe explicitly:
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(a) The algebra generated by C . (b) The σ -algebra generated by C . 5. Let X be a set. An outer measure in X is any map μ : P ( X ) [0 , ] with the following properties. (a) μ ( ) = 0. (b) If A B X , then μ ( A ) μ ( B ). (c) If { A n } is a sequence of subsets of X , then μ ˆ [ n =1 A n ! X n =1 μ ( A n ) . Suppose we define for A R , μ ( A ) = max(sup A, 0). Is μ an outer measure in R ? You should use here, incidentally, that sup = -∞ and that a set is bounded above if and only if its sup is < ; in other words, the sup of sets that are not bounded above is . Solution. Solution. We prove all the properties hold. (a) μ ( ) = max(sup , 0) = max( -∞ , 0) = 0.
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