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I was reading the papers co-authored by Harrison, Kreps and Pliska, that initiated the formal research on the connection between pricing, martingale measures, arbitrage and completeness. I have some issues relating some of the ideas, though, and would be happy if somebody could help me.

In a more or less abstract setting, one considers an ordered topological space $X$ of cash flows, a subspace $M\subseteq X$ of traded instruments and a positive linear functional $\pi:M\to \Bbb R$. The 1st FTAP that relates absence of arbitrage to the existence of EMM concerns continuous extension of $\pi$ to $X$ which is still positive. As an example, one can consider $$ M = \left\{m = m_0 + \int_0^T H_t\;\mathrm dS_t \right\} $$ where $H$ is some "good" strategy and $S$ is a vector of traded assets. In such case, one defines $\pi(m) = m_0$ as an initial cash endowment required to get $m$ at the maturity time $T$.

I guess, the existence of a unique positive extension of $\pi$ to the whole $X$ implies the completeness of the market: that is, we can price everything in a consistent way and there is a unique way to do this. Apparently, this automatically leads to the representability of any $x\in X$ as $x_0 + \int_0^T H_t \mathrm dS_t$.

At the same time, I do not understand, how the existence of at least one continuous positive extension may be related to the absence of arbitrage. That is, I would imagine an arbitrage being rather existence of two strategies $H'$ and $H''$ such that $$ m'_0 + \int_0^T H'_t\;\mathrm dS_t = m''_0 + \int_0^T H''_t\;\mathrm dS_t $$ with $m'_0 \neq m''_0$. But this rather says $\pi$ is not uniquely defined even on $M$ instead of it is not possible to extend $\pi$ to $X$. Does that mean that the latter two statements are equivalent?

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