Measure of a Subset of [0, 1]
Problem Statement: Compute the measure of obtained by deleting all elements with a digit 4 appearing in their decimal expansions.
Solution Attempt: The first thing I noticed is that this very similar to a Cantor set, where the subset is formed by deleting all elements with the digit 2 appearing in their ternary expansions. This leads me to believe (although it seems unintuitive) that the set has measure zero (although I could be wrong, since I just finished an exercise on "Fat" Cantor sets).
In the first stage of the construction, we remove the interval .
In the second stage, we remove the intervals and so on...
So it seems that by the kth stage, what we have left is intervals of length which implies that the measure is zero by taking the limit.
But is this rigorous? It seems similar to the cantor set where you get .
I'd like to also be able to show that the limit of the total length removed is equal to one as well.
Any help would be appreciated - thanks! And please...no probabilistic arguments (I'm not at that level yet!)
Re: Measure of a Subset of [0, 1]
Well, I thought about it more and here is my solution:
We will construct a subset as follows: At each kth step, restrict the subset to values which do not have a digit 4 appearing in the kth digit slot of their respective decimal expansions. Thus, after k steps, what remains is segments of length . On the other hand, at each individual jth step, we remove segments of length , and therefore by the kth step we have removed a total of segments.
We condense these facts into the two following observations in the limit as :
(Total length remaining).
(Total length removed).
We conclude therefore that .
Indeed, it is evident this is true for any similar set since where d = # of digits in chosen expansion (binary, ternary, decimal, etc.).