Proof
Let be the standard mollifier.
Fix a compact set and put be the distance between and the boundary of .
For each and the function
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belongs to test functions and so we may consider
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We assert that it is independent of . To prove it we calculate for .
Recall that
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where the standard mollifier kernel on was defined at Mollifier#Concrete_example. If we put
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then .
Clearly satisfies for . Now calculate
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Put so that
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In terms of we get
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and if we set
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then with for , and . Consequently
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and so , where . Observe that , and
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Here is supported in , and so by assumption
- .
Now by considering difference quotients we see that
- .
Indeed, for we have
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in with respect to , provided and (since we may differentiate both sides with respect to . But then , and so for all , where . Now let . Then, by the usual trick when convolving distributions with test functions,
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and so for we have
- .
Hence, as in as , we get
- .
Consequently , and since was arbitrary, we are done.