I want to show that if O is collection of open subsets of (0,1) what is the closure of O in the associated metric space of equivalence classes? The metric associated with this collection is pseudometric which is equal to outer measure of symmetric difference of two subsets of (0,1).
question related to outer measure and pseudometric.
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analysis
1 Answers
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Hint: For any $E \subset (0,1)$ and $\varepsilon>0$ by definition of outer measure we can find an $O$ such that $E \subset O$ and $m^\ast(O \setminus E) < \varepsilon$.
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0I thought on these lines but infact I want to make a sequence of open sets that converges to limit in (0,1), as I want to show that this collection is complete or not? – 2012-11-13
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0Well, if you prove my hint then you've shown that the open sets are dense. If they're dense, then what does their closure have to be? – 2012-11-13
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0well Jacob, I started with the sequence of open sets contained in (0,1) and constructed the sequence in such a way that it goes with the conditions of nested theorm, now my only concern is that can I show this sequence to be a convergent sequence? If so then I can go to the fact of completeness through density and closure definitely. What's your opinion ? – 2012-11-14