Consider a possible finite sequence $\xi_1,\xi_2,\dots$ of random variables and consider the measure-valued empirical process $$\eta_n=\frac{\sum_{i=1}^n\delta_{\xi_i}}{n},\:\:\: n=1,2,\dots$$ Assume $\eta_n$ is a reverse martingale, in the sense that $(\int fd\eta_n)$ is a reverse-martingale for every $f\geq 0$. Does it automatically hold that $(\eta_n)$ is exchangeable? I.e., does it hold that for any $f_1,\dots,f_n$ and permutation $p$ of $\{1,\dots,n\}$
$$E\left(\int f_1\eta_1\cdots \int f_n\eta_n\right )= E\left(\int f_1\eta_{p(1)}\cdots \int f_n\eta_{p(n)}\right)?$$
I've managed to prove that the last two terms commute, by an argument along
\begin{eqnarray*} E\left(\int f_1\eta_1\cdots \int f_n\eta_n\right)&=&E\left(E\left(\int f_1\eta_1\cdots \int f_n\eta_n|\mathcal{F}_{n}\right)\right)\\ &=&E\left(\int f_1\eta_1\cdots E\left(\int f_{n-1}\eta_{n-1}\int f_n\eta_n|\mathcal{F}_{n}\right)\right)\\ &=&E\left(\int f_1\eta_1\cdots \int f_{n-2}\eta_{n-2}\int f_{n-1}\eta_{n}\int f_n\eta_n\right), \end{eqnarray*} where the reverse-martingale property was used. Here $\mathcal{F}_n=\sigma(\eta_n,\eta_{n+1},\dots)$. However, this argument does not seem to extend to more indices. I'm beginning to suspect there may be a counter-example where exchangeability does not hold.
Any input or reference is appreciated.
I posted this question also on MSE
