= Continuous approximation of Borel indicators on a compact metric space
{title2=$\|\varphi-\mathbf1_B\|_{L^1(\mu)}<\epsilon,\quad0\leq\varphi\leq1$}
For a <Borel probability measure> on a <compact metric space>, <inner regularity> and <outer regularity> give $K\subseteq B\subseteq U$ with $K$ compact, $U$ open and $\mu(U\setminus K)<\epsilon$. The <Urysohn lemma> supplies a <continuous function> $\varphi$ equal to one on $K$ and zero outside $U$, with values in $[0,1]$. Its difference from the <indicator function> of $B$ is supported in $U\setminus K$ and bounded by one, proving the displayed estimate. This transfers identities established for continuous test functions to measurable sets.
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