= Singular support of an indicator function
{title2=$\operatorname{sing\,supp}\mathbf1_A=\operatorname{esscl}_\Omega A\cap\operatorname{esscl}_\Omega(\Omega\setminus A)$}
For any <Lebesgue measurable set> $A\subseteq\Omega$, the <singular support> of its <indicator function>, viewed as a <regular distribution>, is the intersection of the two <essential closures> in the displayed formula. Outside this intersection the <indicator function> is constant <almost everywhere> on a ball and hence represents a <smooth function>. At a point in the intersection, each sufficiently small ball has positive <Lebesgue measure> in both sets. Any continuous representative on that ball must take only the values zero and one everywhere: every other value would persist in a smaller ball and violate equality <almost everywhere>. By connectedness of the ball it would then be constant, a contradiction.
If $A$ is open, its <essential closure> is its relative closure. Its <singular support> can still be smaller than its topological <boundary>: $A=\mathbb R^n\setminus\{0\}$ represents the constant one and has empty <singular support>.
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Target: Support and singular support rules for convolution. Parent: Convolution of distributions with a compactly supported factor. Intended file: fourier-analysis.bigb.
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