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.
By taking the complementary, showing that the supports are equal is equivalent to show that the biggest annihilating sets are equal.
So it is enough to show that an annihilating set for $\mu$ is an annihilating set for $T_{\mu}$ and conversly. That was the suggestion of your professor.