= Complete markets have unique state-price densities
In a one-period <complete market> with a strictly positive <numéraire>, at most one positive <state-price density> prices every traded asset. Suppose $\rho,\rho'$ do so. For each event $A$, replicate the <contingent claim> $S_1^{(0)}\mathbf1_A$. Linearity of the pricing equations gives
$$
\mathbb E[(\rho-\rho')S_1^{(0)}\mathbf1_A]=0.
$$
Both weighted densities are integrable because they price the <numéraire>. An integrable random variable whose integral over every event is zero vanishes almost surely: take the events where it is positive and negative. Therefore $(\rho-\rho')S_1^{(0)}=0$, and strict positivity of the <numéraire> gives $\rho=\rho'$. This argument avoids assuming that the unweighted densities themselves are integrable.
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