Lebesgue measure of numbers whose decimal representation contains at least one digit equal $9$ Let $A$ be the set of numbers on $[0, 1]$ whose decimal representation contains at least one digit equal $9$. What is its Lebesgue measure $\lambda(A)$?

We don't care about the countable set of $x\in[0,1]$ possessing two different decimal expansions. Denote by $A_k$ $(0\leq k\leq9)$ the set of numbers in $[0,1]$ having $k$ as first digit after the decimal point. The sets $A_k$ with $k<9$ can be written as  $$A_k={k\over10}+{1\over 10}A\qquad(0\leq k<9)\ .$$ The scaling property of Lebesgue measure $\lambda$ then implies that $\lambda(A_k)={1\over10}\lambda(A)$ for these $k$. In this way we obtain
$$\lambda(A)=9\cdot{1\over10}\lambda(A)+{1\over 10}\ ,$$
whereby the last term measures the set $A_9$.
It follows that $\lambda(A)=1$.
