Example 2.4.2E2DD6B809EB8

Example 2.4.2.E2DD6B809EB8 For $x\in \Omega $, the Dirac measure $\delta _ x$ assigns $1$ to Borel sets containing $x$ and $0$ to the others; as a distribution, $\langle \delta _ x,\chi \rangle =\chi (x)$. In one real dimension, with $0\in \Omega $, we have $\langle \delta '_0,\chi \rangle =-\chi '(0)$. This distribution is not given by integration against a real signed Radon measure in the sense of section 2.2. Indeed, let $\theta $ be a test function with $\theta (0)=1$, for instance a multiple of the function $\rho _\varepsilon $ of section 2.5 with $\varepsilon $ small. The test functions $\theta (x)\sin (nx)$ are bounded by $\sup |\theta |$ and supported in $\operatorname{Supp}\theta $, while their pairings with $\delta '_0$ equal $-\theta (0)n\cos 0-\theta '(0)\sin 0=-n$. Integration against a real signed Radon measure $\sigma $ would bound these pairings by $\sup |\theta |\cdot |\sigma |(\operatorname{Supp}\theta )$, since $|\int f\, \mathrm d\sigma |\leq \int |f|\, \mathrm d|\sigma |$.