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Nilanjan Bhowmick AIR 3, CSIR NET (Earth Science)
Shreya Desai posted an Question
February 28, 2020 • 15:15 pm
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30 points
30 points
- CSIR NET
- Mathematical Sciences
How to prove that the outer measure of cantor set is zero
how to prove that the outer measure of cantor set is zero
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Krishna
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Consider a closed and bounded set FF in the open interval (−n,n)(−n,n) of RR. Then in the usual topology with Lebesgue measure μ(F)=2n−μ((−n,n)∖F)μ(F)=2n−μ((−n,n)∖F). The Cantor set satisfies this property so it can be our FF. That is, CC is in [0,1][0,1] and can also be said to be in [−1,1][−1,1]. CC is bounded because it has a supremum and infimum, in this case 11 and −1−1. Write (−1,1)∖C=(−1,0)∪[0,1]∖C(−1,1)∖C=(−1,0)∪[0,1]∖C. So μ((−1,1)∖C)=1+μ([0,1]∖C)μ((−1,1)∖C)=1+μ([0,1]∖C), and this apparently implies CC has measure zero.
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can you be more specific about outer measure?
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Sourav ghosh
Cantor set is the intersection of sequence of non increasing closed intervals. Outer measure of Cantor set is 0 because it the Inf of sum of the length of intervals and the n th interval contains 2^n intervals each of length 1/3^n . So, the series (2^n)/(3^n), is a gp series with common ratio <1. Converges to 0 and so outer measure of Cantor set is <= 0. since outer measure is always non negative, so outer measure of Cantor set is 0.