The measure of a Cantor Set

Question

So I am trying to argue that the set of points in $[0,1]$ which will not have $4$ in their decimal expansion has measure $0$. So I am thinking to construct a step function such that this set is a null set hence it will have measure $0$. But I am kind stuck in constructing such step functions, any hints?

Answer 1

HINT: $$\begin{align*} \frac1{10}+\frac1{10}\cdot\frac9{10}+\frac1{10}\left(\frac9{10}\right)^2+\ldots&=\frac1{10}\sum_{n\ge 0}\left(\frac9{10}\right)^n\\ &=\frac1{10}\cdot\frac1{1-\frac9{10}}\\ &=1 \end{align*}$$