 Solution

On a finite sample space, all real-valued holdings over the finite interval $0,\dots ,T$ are bounded after null states are discarded. The gains identity is therefore a bounded predictable martingale transform of the vector martingale $P$, summed over its coordinates. It follows that $X$ is a martingale, soHere the replication cost is a prescribed deterministic initial capital, as in the definition of attainability. The stronger intermediate identity is $X_t=\mathbb{E}[\xi \mid {\mathcal{F}}_t]$. If initial capital is instead allowed to be ${\mathcal{F}}_0$-measurable and random, the corresponding statement is $X_0=\mathbb{E}[\xi \mid {\mathcal{F}}_0]$; its unconditional expectation still equals $\mathbb{E}\xi$.