Hello.

Problem Statement: Compute the measure of $\displaystyle E\subset [0, 1]$ 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$\displaystyle I = (0.4, 0.5]$.

In the second stage, we remove the intervals $\displaystyle I = (0.04, 0.05]; (0.14, 0.15]; (0.24, 0.25]; (0.34, 0.35]$ and so on...

So it seems that by the kth stage, what we have left is $\displaystyle 9^{k}$ intervals of length $\displaystyle 10^{-k}$ 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 $\displaystyle (\frac{2}{3})^{k} \rightarrow 0$.

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!)