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 .
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.