# Question about a distribution definition in $D'(\Omega)$

So, let $$\Omega :=B_1(0) \subset R^3$$ and $$\{x_n\} \subset \Omega$$. If $$x_n \rightarrow x \in \partial B_1(0)$$, $$T:= \sum_{n \in N}\delta_{x_n}$$ define a distribution on $$D'(\Omega)$$? How can I check it?

One idea of mine was to check if there is a function $$f$$ so that $$T_f$$ can be identified with the function (with an abuse of notation).

Using the definition I'll have:

$$\langle T_f,\phi \rangle = \int_{- \infty}^{+ \infty} \sum_{n \in N}\delta_{x_n} \phi dx=\sum_{n \in N}\int_{- \infty}^{+ \infty} \delta_{x_n} \phi dx$$

for $$\phi \in C_c^\infty(\Omega)$$

But here I don't know how to formalize the check. Any tips?

• Remember that $\phi$ has compact support inside of $\Omega$. Therefore only a finite number of $x_n$ will be inside the support of $\phi$ and actually be used in the sum. – md2perpe Oct 19 at 16:04
• Continuity of $T$ on $\mathscr D(\Omega)$ means, by definition, that for every compact set $K$ the restriction of $T$ to $\mathscr D(K)$ is continuous. – Jochen Oct 20 at 16:07
• Ok, but I have difficulties to apply it to this exercise; it means that it defines a function and my procedure is correct? – Alessar Oct 24 at 11:56