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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 $f \in \mathcal{D}(\mathbb{R}_{+})$?

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Yes, this is true. The proof is long (longer than I am willing to write up unfortunately) and is virtually identical to the proof of the Riesz Representation Theorem if you know that one. You can find this in Lieb and Loss 'Analysis' thm 6.22. – Chris Janjigian Nov 20 '12 at 18:50
@Chris thank you very much! – Nimza Nov 20 '12 at 20:40

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