Let $I$ be a compact interval and $\mathcal{M}(I)$ the space of (signed) Borel measures. We equip it with the weak topology, i.e. a sequence $\mu_n$ converges to zero if and only if $$ \left|\int_I f(x) \mathrm{d}\mu_n(x)\right| \longrightarrow 0$$ for all $f \in C(I)$.

**Now the question is the following: Let $V \subset \mathcal{M}(I)$ be the vectorspace of all finite linear combinations of Dirac measures supported at different points in $I$. Is $V$ dense in $\mathcal{M}(I)$?**

For example if $I = [0,1]$, the sequence $$ \mu_n = \frac{1}{N}\sum_{j=1}^N \delta_{j/N},$$ $\delta_{j/N}$ being the Dirac measure supported at $j/N$, weak*-converges to the Lebesgue measure as $\mu_n$ is just the approximation by Riemann sums. Hence one can easily get all measures that are absolutely continuous w.r.t. the Lebesgue measure.

However, there are more measures (singular measures) that are neither point measures nor Lebesgue measures and I don't have an idea how to reach those.