The question is based on a one period model. Let a market be arbitrage free, and then let a security $X$ be added to it. Denote $P(X)$ as the price of this security at $t=0$. The security has the property that:
$$A=\text{inf}\{E(\phi X)|E(\phi X)<\infty,\phi \in \Phi\}$$ $$B=\text{sup}\{E(\phi X)|E(\phi X)<\infty,\phi \in \Phi\}$$
$A<B$, $P(X)\in (A,B)$. Where $\Phi$ is the set of state space deflators. I want to show that the market is arbitrage free.
Any tips on how to prove this result? And what is $E(X)$ even supposed to mean here? I suppose it's the expectation of the price on period 1?
Attempt ($Q$ denotes quantity of a specific security. $S(t)$ the price at $t$):
Let value of a portfolio: $\sum^n_{i=1}Q_i S_i(0)=P(X)=E(\phi X)$
Therefore there exists a deflators $\phi_1,\phi_2$:
$$E(X\phi_1)\leq\sum^n_{i=1}Q_iE(\phi_1 S_i(0))=E(\phi X)=\sum^n_{i=1}Q_iE(\phi_2 S_i(0))\leq E(X\phi_2)$$
Now assume that in every case:
$$\sum^n_{i=1}Q_i S_i(1)\geq X$$
And there is a possibility: $$\sum^n_{i=1}Q_i S_i(1)> X$$
This is impossible by the above, but how better show it?