The Smith-Volterra-Cantor set SVC or the so-called "Fat Cantor set" is a nowhere dense set with a positive Lebesgue measure.
The following is a question from A radical approach to Lebesgue’s theory of integration by Bressoud Q4.4.6 pg 118.
Define K1 to be the fat Cantor set and K2 to be the subset of [0,1] formed by putting a copy of K1 inside every open interval in [0,1]∖K1. Similarly define K3 to be the subset of [0,1] formed by putting a copy of K1 inside every open interval in [0,1]∖(K1∪K2). In general define Kn to be the subset of [0,1] formed by putting a copy of K1 inside every open interval in [0,1]∖(K1∪K2∪…∪Kn−1). Define K=∞⋃n=1Kn. Show that K is of full measure.
Q1. What does it mean to put a copy of K1 inside every open interval in [0,1]∖K1?
My understanding:- The open interval (38,58)∈[0,1]∖K1. I guess to put a copy of K1 inside it is to repeat the process by which we got the fat Cantor set from [0,1] on (38,58) with shorter intervals being removed now. Am I correct?
Q2. How to show that K is of full measure?
Thanks!
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