The market is arbitrage-free iff there exists an equivalent martingale measure for the discounted price process of the stock.
So in a world with a finite amount of possible outcomes $\Omega$ that follow the probability distribution $P$, where we can change $P$ by an equivalent probability measure that is a martingale, there are no arbitrage opportunities.
Now this martingale measure is supposed to model a 'fair game/price'. Could someone please shed some light on why a 'fair price' implies there is no arbitrage? What is a 'fair price'?