On a Lemma of Schachermayer

In this paper we prove a topological lemma on real valued random variables which implies the basic ingredients for the proof of the Fundamental Theorem of Asset Pricing in the two period case. In particular, previous results of Stricker, 9], and of Schachermayer, 6], are special cases of our result. Our proof is considerably shorter and more transparent than previous proofs of related special cases. Let L 0 ((; A; P 0) be the space of real valued random variables endowed with the topology of convergence in P 0-measure and let B be a sub-eld. The main result is concerned with B-modules, i.e. subspaces which are modules for the ring of B-measurable random variables. Let C L 0 be a cone which is closed under multiplication with nonnegative B-measurable functions. Let K and L be B-modules and assume that K is nitely generated. Our main result asserts: If L ? C is closed and if (K + L) \ C = f0g then K + L ? C is closed, too.