A note on extremality and completeness in financial markets with infinitely many risky assets

In this paper we study the interplay existing between completeness of financial markets with infinitely many risky assets and extremality of equivalent martingale measures. In particular, we obtain a version of the Douglas- Naimark Theorem for a dual system ‹X, Y› of locally convex topological real vector spaces equipped with the weak topology σ(X, Y), and we apply it to the space L∞ with the topology σ(L∞, Lp) for p≥1. Thanks to these results, we obtain a condition equivalent to the market completeness and based on the notion of extremality of measures, which allows us to give new and simpler proofs of the second fundamental theorems of asset pricing. Finally, we discuss also the completeness of a slight generalization of the Artzner and Heath example.