Linear combinations of delta measures

Solution 1:

You can also "just" apply Hahn-Banach. The dual space of $(C_0(\mathbb R)^*,w^*)$ is $C_0(\mathbb R)$. So, to prove that the linear combinations of Delta measures are $w^*$-dense it is enough to check that if a function $f\in C_0(\mathbb R)$ is such that $\int f d\delta_a=0$ for every $a\in\mathbb R$ then $f=0$; but this is obvious since $\int f d\delta_a=f(a)$.

Solution 2:

Dirac measres are extreme points of the convex set of subprobability measures $P\subset C_0(\mathbb{R})^*$. This set is compact in the weak-$^*$ topology, so by Krein-Milman theorem every probability measure is a weak-$^*$ limit of finite convex combination of Dirac delta measures. For details see example 8.16 here.