= Solution
In the one-period model, initial holdings are a deterministic vector $H\in\mathbb R^n$, chosen with trivial initial information, and the terminal payoff of the portfolio is $H^TP_1$. A <contingent claim> $\xi_1$ is replicable if some such vector has $H^TP_1=\xi_1$ almost surely; its initial <claim replication> cost is $H^TP_0$. A <complete market> replicates every claim in the specified terminal-information class; for the arguments below, it suffices that every bounded measurable claim is replicable. The $n$ assets count all assets of the model, including a cash asset if one is traded.
These are one-period definitions. If arbitrary terminal information were already available initially and holdings could depend on it, the finite-atomic conclusion in the next part would not follow; the deterministic-holdings convention is essential.
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