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I need a reference on where to find a modified version of Hahn-Banach Theorem that ensures that if $F(g) = \int g d\mu$ defines a positive linear functional on a subspace of $C_{0}(M, \mathbb{R})$, $F$ extends to $C_{0}(M, \mathbb{R})$, still positive.

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    $\begingroup$ Theorem 2.1 spot.colorado.edu/~baggett/funcchap2.pdf $\endgroup$ – Tomek Kania Sep 9 at 21:39
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    $\begingroup$ The OP's modified version isn't true as it is stated. E.g., think about $C[0,1]\supset\{\alpha t\sin(1/t)+\beta t^2 : \alpha,\beta\in {\bf R}\}\ni\alpha t\sin(1/t)+\beta t^2\mapsto \alpha$. Some assumption has to be made on the subspace where $F$ is defined. A standard assumption is found in Kania's post. $\endgroup$ – Narutaka OZAWA Sep 10 at 3:59
  • $\begingroup$ Space M is $\Sigma = \{ (x_n)_n: n \in \mathbb{Z} \}$ where the symbols belongs to $\{ 1, \cdots, m\}$ and exists a matrix of transition $A$ such that $A_{x_n, x_{n+1}}=1$. I believe it is possible to get a cone in this space and use the theorem that @TomekKania posted. Thank you all for your help! $\endgroup$ – Ricardo Freire Sep 10 at 22:16

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