On the Second Fundamental Theorem of Asset Pricing

Let $X^1,\dots ,X^d$ be sigma-martingales on $(\mathrm{\Omega },\mathcal{F},P)$. We show that every bounded martingale (with respect to the underlying filtration) admits an integral representation w.r.t. $X^1,\dots ,X^d$ if and only if there is no equivalent probability measure (other than $P$) under which $X^1,\dots ,X^d$ are sigma-martingales. From this we deduce the second fundamental theorem of asset pricing- that completeness of a market is equivalent to uniqueness of Equivalent Sigma-Martingale Measure (ESMM).