# Support of a Distribution

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.

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.
• For a distribution $T$ an annihilating set $U$ is a set for which $T(\varphi) =0$ for all $\varphi$ supported by $U$. The support of $T$ is by definition the complementary of the biggest annihilating set. (I.e the union of all annihilating sets) May 19, 2016 at 17:53
• So I need to show then that the biggest open set $U$ s.t. $T(\phi) = 0$ with $supp(\phi) \subset U$ is the same $U$ s.t. U is the union of all open sets $N$ with $\mu(N) = 0$? May 19, 2016 at 18:04