Stochastic process M_t adapted to filtration mathcalF_t with mathbbE[M_t]<∞ and mathbbE[M_t,|,mathcalF_s] = M_s for all s ≤ t — a fair game. The fundamental theorem of asset pricing says no-arbitrage is equivalent to existence of an equivalent measure under which discounted prices are martingales (FTAP-I); market completeness is uniqueness of that measure (FTAP-II). Itô integrals of L^2-integrands against Brownian motion are martingales by construction.