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Let $\Omega \subset \mathbb{R}^N$ be open, and consider a distribution $T\in \mathcal{D}’(\Omega)$. I should prove the following statement.

The distribution $T$ is a linear combination of Radon measure on $\Omega$ iff the following property is true:

For all $K\subset \Omega$ compact, there exists a constant $C_K$ such that $|\left<T,\phi\right>| \le C_k \|\phi\|_\infty$ for all $\phi \in C^\infty_c(\Omega)$ with $\operatorname{supp}\phi \in K$.

It’s very easy to prove the direction $\Rightarrow$, but I’m having some problems with the other direction. I think I should use the Hahn-Banach Theorem and the Riesz-Markov-Kakutani representation theorem. I’m writing my ideas in the comment. Can someone, please, help me?

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  • $\begingroup$ This is the version of the R-M-K representation theorem I know: * If $L: C_c(\Omega) \to \mathbb{ K}$, with $\mathbb{ K}$= \mathbb{ R}$ \quad or \quad \mathbb{ C}$ such that for all $K\subset \Omega$ compact there exists a constant $C_K$ with $|<T,\phi> \le C_K ||\phi||_\infty$ for all $\phi \in C_c(\Omega)$ with $ supp\phi \in K$, then $T$ is a linear combination of functional of the form $\int \phi d\mu$ with $\mu$ Radon measure* $\endgroup$
    – Shiva
    May 15 at 8:09
  • $\begingroup$ I think I can use this theorem provided passing from $C^\infty_c(\Omega) to C_c(\Omega)$. To do it I would use the Hahn Banach theorem in the following way: $\endgroup$
    – Shiva
    May 15 at 8:11
  • $\begingroup$ I define the functional $T$ just on $C_c^\infty (\Omega)$ which is a linear subspace of $C_c(\Omega)$, but the problema is that the property in the statement of the theorem does not guarantee me that my distribution T is a continuous function on $C^\infty_c(\Omega)$ (it is continuous but in sense of distribution). $\endgroup$
    – Shiva
    May 15 at 8:15

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