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?
2026-04-04 10:59:49.1775300389
Derivative of Dirac delta function as a measure
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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.