= Essential closure
{title2=$\operatorname{esscl}_\Omega A$}
For a <Lebesgue measurable set> $A$ in an <open set> $\Omega\subseteq\mathbb R^n$, its essential closure consists of the points whose every sufficiently small ball has positive <Lebesgue measure> intersection with $A$. Equivalently,
$$
\operatorname{esscl}_\Omega A=\operatorname{supp}\mathbf1_A,
$$
where the right side is the <support of a distribution>. To see the equivalence, a ball on which $A$ is null gives zero pairings with all <test functions>. If $A$ has positive measure in a ball, cover that ball by smaller compactly contained balls and choose a nonnegative <cutoff function> positive on one meeting $A$ in positive measure; its pairing is positive. Thus essential closure is relatively closed and unchanged by modifying $A$ on a <Lebesgue null set>. For an <open set> $A$ it equals its relative closure.
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Target: Singular support of an indicator function. Parent: Singular support. Intended file: distribution-theory.bigb.
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