Uniqueness property for the space of finite measure. Let $\mu$ be a finite measure on $\mathbb{R}$ satisfying,
$$\int_{\mathbb{R}}f(x)d\mu(x)=0,~\forall f\in C_c(\mathbb{R})$$
Then is it true that $\mu =0$?
We know that the result is true for $L^1(\mathbb{R})$, which is a subspace of the above.
Edit after the comments of Kavi Rama Murthy Sir:
Is the result also true for complex measure $\mu$ on $\mathbb{R}$?

For all $n=1,2,3,\dots$ there exists a non-negative function $f \in \mathrm{C}_{\mathrm{c}}(\mathbb{R})$ such that $f=1$ on $[-n,n]$. Therefore, for all $n=1,2,3,\dots$
\begin{equation}
\mu([-n,n]) \leq \int_\mathbb{R} f d \mu = 0.
\end{equation}
By countable subadditivity $\mu(\mathbb{R})=0$, and we are done.
