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(Solved): Construction of the Cantor Set Let (\alpha _(n))_(ninN) be a sequence of real numbers such that \alp ...
Construction of the Cantor Set Let
(\alpha _(n))_(ninN)be a sequence of real numbers such that
\alpha _(n)>0for all
nand
\sum_(n=0)^(\infty ) 2^(n)\alpha _(n)<=1We construct a decreasing sequence of compact sets
(A_(n))_(ninN)within
0,1as follows: Initial Set: Define
A_(0)=[0,1]. Iterative Construction: Define
A_(1)=[0,(1-\alpha _(0))/(2)]\cup [(1+\alpha _(0))/(2),1].For each
ninN,A_(n+1)is obtained by removing an open, centered interval of length
\alpha _(n)from each of the
2^(n)intervals that make up
A_(n). Questions Show that for each
n,A_(n+1)subA_(n)and that
A_(n)consists of
2^(n)disjoint intervals of equal length. 1 Prove that the length of each interval in
A_(n)is
(1-\sum_(k=0)^(n-1) 2^(k)\alpha _(k))/(2^(n)). Demonstrate that the total measure of
A_(n)is given by
\lambda (A_(n))=1-\sum_(k=0)^(n-1) 2^(k)\alpha _(k)Solv all the questions step by step and expline for me how you solve it by detileals
