Saturday, 5 August 2017

real analysis - Making a dense set of full measure from Cantor like sets

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](K1K2). 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](K1K2Kn1). 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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