Consider the delta dirac distribution $\delta (\varphi) = \varphi (0), \varphi \in \mathcal{S}(\mathbb{R}^n)$ (the Schwartz space). I know that $\delta ^{'} (\varphi) = - {\varphi }^{'} (0)$. How can I prove $\delta^{'}$ is not given by a measure, that is , doesn't exists a measure $\mu$ such that

$$\delta^{'} (\varphi) = \displaystyle\int_{R^n} \varphi (x)\,d \mu (x) $$

I have no idea how to proceed. Someone can give me a hint ?

Thanks in advance


Choose a test function $\varphi$ with $\frac{\partial \varphi}{\partial x_i}(0) = 1$. For $n \in \mathbb{Z}^+$, let

$$f_n(x) = \frac1n\varphi\left(nx\right).$$

For every measure $\mu$, we have

$$\lim_{n\to\infty} \int f_n(x)\,d\mu = 0$$

by the dominated convergence theorem. But

$$\frac{\partial\delta}{\partial x_i}[f_n] = - \frac{\partial f_n}{\partial x_i}(0) = -1$$

for all $n$.

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  • $\begingroup$ Is trivial to give a explicit form to your test function ? $\endgroup$ – math student Nov 6 '13 at 0:05
  • $\begingroup$ You could take $\varphi(x) = x_i\cdot \psi(x)$ where $\psi$ is a cutoff function, say $\psi(x) = \psi_0(\lVert x\rVert)$ and $\psi_0\colon \mathbb{R}\to\mathbb{R}$ is a function with $\psi_0(x) = 1$ for $x \leqslant 1$, $\psi_0(x) = 0$ for $x \geqslant 2$ constructed using $\exp \left(-\frac{1}{x^2}\right)$. I wouldn't call it trivial, but you can construct an explicit example. $\endgroup$ – Daniel Fischer Nov 6 '13 at 0:12
  • $\begingroup$ I understand that the sequence $f_n$ is dominated by $\frac{1}{n} || \varphi||_{\infty}$. But I have dominate by a Integrable function . Which integrable functions do you take ? (my english is terrible, sorry) $\endgroup$ – math student Nov 6 '13 at 0:23
  • $\begingroup$ Take a cutoff function $\eta$ that is $\equiv 1$ on a ball containing the support of $\varphi$. Since $\operatorname{supp} f_n = \frac1n\operatorname{supp}\varphi$, the sequence $f_n$ is dominated by $\lVert\varphi\rVert_\infty\cdot \eta$. $\endgroup$ – Daniel Fischer Nov 6 '13 at 0:26
  • $\begingroup$ I understand your construction of the test function , your last coment and the solution for my problem . Thank you !!! . Nice solution for my problem! $\endgroup$ – math student Nov 6 '13 at 1:02

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