Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
2  
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.