Let $(\Omega,\mathcal F,P)$ be a probability space, and let $X:=L^p$ denote the normed space of (equivalence classes) of $p$-integrable real random variables on $(\Omega,\mathcal F,P)$, where $1\leq p<\infty$.
Definition. Let $M$ be a subspace of $L^p$ and $\pi:M\to \mathbb R$ be a linear functional on $M$. We say that $(M,\pi)$ has the extension property if there exists a strictly positive continuous linear functional $\psi:X\to \mathbb R$ such that $\psi|_M=\pi$.
Strictly positive means $\psi(x)>0$ if $x\in L_+^p\setminus\{0\}$, where $L_+^p=\{x\in L^p : x\geq 0\}$ .
Let $M_0=\{m\in M: \pi(m)=0\}$ and let $C=M_0-L_+^p$. Suppose the no arbitrage condition $$\bar{C}\cap L_+^p=\{0\}$$ holds, where $\bar{C}$ denotes the closure of $C$. It follows from the Kreps-Yan theorem that there exists a strictly positive continuous linear functional $g:X\to \mathbb R$ such that $g|_C\leq 0$.
How can I use this to show that $(M,\pi)$ has the extension property?
This claim is made right after Theorem 1.4 here.
Thanks a lot for your help.
Edit:
The condition $g|_C\leq 0$ implies that the null space of $\pi$ is a subspace of the null space of $g|_M$. Using this result we get that $g|_M=\lambda \pi$ for some $\lambda\in\mathbb R$.
If $\lambda>0$ then $\psi=\frac{1}{\lambda}g$ gives the desired extension. But how to deal with the case $\lambda\leq0$?
