Prove that the functional in $C_c^0(Omega)$ is a Radon measure

Let $Omega subset mathbb{R}^n$ be an arbitrary open set and $(x_n)_{n inmathbb{N}} subset Omega$ a sequence. Let $(a_n)_{n inmathbb{N}} subset mathbb{C}$ be a sequence such that
$$sum_{j=1}^{infty} |a_j| < infty.$$

I want to prove that the functional $T:C_c^0(Omega) longrightarrow mathbb{C}$
given by
$$T(varphi)=sum_{j=1}^{infty} a_jvarphi(x_j),; forall ; varphi in C_c^0(Omega)$$
is a Radon meausure in $Omega$, that is, I want to prove that $T$ is continuous when $C_c^0(Omega)$ is equipped with the topology inductive limit of the spaces $C_c^0(K)$, here $K subset Omega$ is an arbitrary compact subset of $Omega$.

I thought of the following: it is enough I prove that $ T $ is continuous in $C_c^0(K)$, for all $K subset Omega$ compact, where the space $C_c^0(K)$ is (in particular) a Banach space with the norm
$$|f|_K=sup_{x in K} |f(x)|, ; forall ; f in C_c^0(K).$$

That’s what I thought true? This is the way?