Martingale characterization by bounded self-financing strategies

For an integrable adapted discounted price process $X$ in finite discrete time, the following are equivalent under a probability measure $Q$: $X$ is a $Q$-martingale; every bounded predictable self-financing strategy has a martingale discounted value; and every such strategy has expected terminal value equal to its initial value. To recover the martingale property from the expectation identity, trade one asset only during one period on an arbitrary event in the preceding sigma-algebra.