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{\bf Question}

Let $S$ be the set of real numbers in $(0,1)$ whose decimal
expansions do not contain the digit 4.  Prove that $m^*(S)=0$.
Generalise this result as far as you can.



\vspace{0.25in}

{\bf Answer}

Consider the interval

$\ds I_{a_1 \cdots
a_n}=\left[\frac{a_1}{10}+\frac{a_2}{10^2}+\cdots
\frac{a_{n-1}}{10^{n-1}}+\frac{a_n}{10^n}, \frac{a_1}{10}+\cdots
\frac{a_n+1}{10^n}\right]=[\alpha,\beta]$

$a_i\epsilon T$ where $T={0,1,2,3,5,6,7,8,9}$

Then $\ds \bigcup_{\begin{array}{c}i=1\\ a_i\epsilon T\end{array}}
^nI_{a_1\cdots a_n} \supseteq S$

For $\alpha \leq a_1\cdots a_na_{n+1}\cdots\leq\beta$

Thus $\ds m^*(S)\leq \sum|I_{a_1\cdots
a_n}|=9^n\frac{1}{10^n}<\epsilon$ for $n>n_0$.


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