Representation of distribution by nonnegative measure

Question

Let $T \in \mathcal{D}'(\mathbb{R}_{+})$ be a distribution on $\mathbb{R}_{+}$ such that for any $f \in \mathcal{D}(\mathbb{R}_{+})$, $f \geqslant 0$ we have $$ \langle f, T \rangle \geqslant 0 $$ Is it true that there exists a nonnegative measure $\mu$ with support on $\mathbb{R}_{+}$ such that $$ \langle f,T \rangle = \int\limits_{0}^{\infty} f(x)\,\mu(dx) $$ for any $f \in \mathcal{D}(\mathbb{R}_{+})$?