Before revision:  Under a continuity assumption, we prove using a backward-forward scheme that the absence of arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of consistent price systems.
After revision:  Under a continuity assumption, we prove 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.