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Distribution Theory/Support and singular support

Definition (being zero on open sets):

Let be a smooth manifold, let be open, and let . Let be an open subset. We say that is zero on iff for all we have .

Proposition (distribution is zero on union of opens where it is zero):

Let ( being a smooth manifold), and let . Suppose that is zero on a family of open subsets (). Then is also zero on

.

Proof: Let . Then is a compact subset of . Hence, extract a finite subcover . Then pick a finite partition of unity on of functions that are subordinate to (using that is a compact subset of and convolving an indicator function on that with a mollifier of sufficiently small support), and use linearity of .

Definition (support):

Let , where is an open subset of a smooth manifold. The support of is the set

,

where the union ranges over all open on which is zero.