Dirac delta function can be defined in several ways. I know two definitions. One is as a distribution and the other is as a measure. I found many materials on the derivatives of delta function as a distribution. However, I couldn't find materials dealing derivatives of delta function as a measure. Could someone point me at any materials or explain it?
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In the sense of distributions, if $\phi$ is compactly supported and smooth in $\mathbb R^d$, $$\delta'(\phi)=-\delta(\phi')=-\phi'(0).$$ But, it is not possible to express this as a measure.
detnvvp
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Do you perhaps have a proof of the last claim? Not doubting or saying you're wrong, just curious. – Feb 22 '16 at 07:11
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A proof can actually be found here: http://math.stackexchange.com/questions/553585/about-the-derivative-of-dirac-delta-distribution . – detnvvp Feb 22 '16 at 08:21