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?