The following question arises while reading the following article by G. Lowther: 'Fitting martingales to given marginals'.
First, let us fix the setting: We consider the space of cadlag real valued processes with coordinate process $S$. So \begin{align} \mathrm{D}=\{\text{cadlag functions }\omega:\mathbb{R}_+\rightarrow\mathbb{R}\}\\ S:\mathbb{R}_+\times\mathrm{D}\rightarrow\mathbb{R},(t,\omega)\mapsto S_t(\omega)=\omega(t) \end{align} with natural sigma-algebra and filtration generated by $S$. Fix a finite time-set, say $T=\{ t_1<...<t_m \}$ and suppose the following: There exists a martingale measure $\mathbb{P}_1$ (under which $S$ is a martingale) matching some given marginal distributions up to time $t_{n-1}$ (so that the law of $S$ under $\mathbb{P}_1$ at each time equals some given distribution) and a martingale measure $\mathbb{P}_2$ matching some given marginal distributions at times $t_{n-1}$ and $t_{n}$. Because $S_{t_{n-1}}$ has the same distribution under these both measures, it is possible to join them together to $\mathbb{P}$ at time $t_{n-1}$. Now he states that $\mathbb{P}$ is the unique measure on $(\mathrm{D},\mathcal{F})$ such that \begin{align} \mathbb{E}_\mathbb{P}[AB]=\mathbb{E}_{\mathbb{P}_1}[A \mathbb{E}_{\mathbb{P}_2}[B\vert X_{t_{n-1}}]]= \mathbb{E}_{\mathbb{P}_2}[\mathbb{E}_{\mathbb{P}_1}[A\vert X_{t_{n-1}}]B] \end{align} for bounded random variables $A,B$ for which $A$ is $\mathcal{F}_{t_{n-1}}$-measurable and $B$ is $\sigma(S_t:t\geq t_{n-1})$-measurable. My question now is the following: Why do I need this last line of equations? What does it give? What does it show? Anyone with some ideas?