= Indicator of a restricted-digit decimal set
{title2=$\int_0^1\mathbf1_K(x)\,dx=0$}
Let $K$ consist of numbers whose decimal digits all lie in a fixed set $D\subseteq\{1,\ldots,8\}$ 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.
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