Exercise 8, § 3

Let $n\in \mathbf{N}$ and U be an open set in $\mathbf{R}^n$. A distribution $f\in \mathscr{D}'(U)$ is said to be positive if $\langle f, \varphi \rangle \geqslant 0$ for every $\varphi \in \mathscr{D}(U)$ such that $\varphi \geqslant 0$. Prove that $f$ is positive if and only if $f$ is the distribution associated with a positive measure on U.