  We prove the Fundamental Theorem of Asset Pricing for a discrete time
financial market where trading is subject to proportional transaction cost and
the asset price dynamic is modeled by a family of probability measures,
possibly non-dominated. Using a backward-forward scheme, we show that when the
market consists of a money market account and a single stock, no-arbitrage in a
quasi-sure sense is equivalent to the existence of a suitable family of
consistent price systems. We also show that when the market consists of
multiple dynamically traded assets and satisfies \emph{efficient friction},
strict no-arbitrage in a quasi-sure sense is equivalent to the existence of a
suitable family of strictly consistent price systems.
