= Solution
The <positive state-price density alternative> characterizes one-period <arbitrage>. Interpret $A_0$ as the vector of current asset prices and $A_1$ as their terminal payments. A fixed vector of holdings produces initial cost $x^\top A_0$ and terminal payment $x^\top A_1$. The first alternative is a nonpositive-cost investment with a nonnegative payment and strict improvement either initially or on an event of positive probability. A strictly positive <state-price density> prices every such payment and excludes both forms of <arbitrage>, as proved below.
If there is a traded <bank account> costing one and paying a deterministic $R>0$, the density satisfies $1=R\mathbb E\nu$. Hence $dQ/dP=R\nu$ defines an <equivalent martingale measure> and
$$
\boxed{A_0=\frac1R\mathbb E_Q A_1.}
$$
Conversely this pricing relation under an <equivalent martingale measure> gives $\nu=R^{-1}dQ/dP$. Without such an account, $\nu$ is a pricing density, not necessarily a probability density or even an <integrable random variable> by itself; the stated requirement is <integrability> of $\nu A_1$. This distinction keeps the abstract alternative valid without an unstated asset paying one.
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