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 $f \in \mathcal{D}(\mathbb{R}_{+})$?
Representation of distribution by nonnegative measure
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measure-theory
distribution-theory
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0@Chris thank you very much! – 2012-11-20