Let $U$ be a nonempty open subset of $\mathbb{R}^n$ and $\mu$ be a Radon measure on $U$. Define $$T_\mu(\phi) = \int_U \phi d\mu$$ for all $\phi \in D(U) = C_c^\infty(U)$. Prove that $T_\mu$ is a distribution on $U$ and that $supp(T_\mu) = supp(\mu)$.
I was able to shwo that $T_\mu$ is linear and continuous, and hence is a distribution. I'm having a bit of trouble showing that the supports are equal. My professor gave me a hint which was to look at any open subset $V \subseteq U$ and that $\mu(V) = 0$ iff $T_\mu = 0$ on $V$. I don't quite see how to use this hint to show the two supports are equal.