Compactly supported continuous functions determine a finite Borel measure
= Compactly supported continuous functions determine a finite Borel measure
If finite <Borel measures> $\mu$ and $\nu$ on $\mathbb R$ satisfy
$$
\int f\,d\mu=\int f\,d\nu
$$
for every compactly supported <continuous function> $f$, then $\mu=\nu$. This follows from the uniqueness part of the <Riesz representation theorem>; alternatively, continuous functions that approximate indicators from above and below recover the measure of every interval and hence every Borel set.