Indicator of a restricted-digit decimal set (${\int }_0^1{1}_K(x)dx=0$)

Let $K$ consist of numbers whose decimal digits all lie in a fixed set $D\subseteq \{1,\dots ,8\}$ with $m<10$ digits. The $m^N$ allowed prefixes give closed decimal intervals of length $10^{-N}$ covering $K$, with no endpoints in $K$. A partition of an interval at those endpoints gives an upper Darboux sum at most $(m/10{)}^N$. Every cell contains a terminating decimal outside $K$, so the lower Darboux sum is zero. Thus the indicator function is Riemann integrable with Riemann integral zero. Such sets illustrate how an uncountable set may have an integrable indicator function of zero integral.