Let $X$ be a countable set. A lottery on $X$ is a function $\lambda: X \to [0,1]$ such that $\sum_x \lambda(x) = 1$. Let $\Delta X$ be the set of lotteries on $X$.
A total preorder $\preceq$ on $\Delta X$ is a transitive, reflexive, complete binary relation on $\Delta X$.
Let $(I, \mathcal I)$ be a measurable space. Note that if $\lambda_i$ is a family of lotteries on $X$ indexed by $I$ and $p$ is a probability measure on $(I, \mathcal I)$, then the function $\int\lambda_ip(di)$ on $X$ defined by $\big(\int\lambda_ip(di)\big)(x) = \int \lambda_i(x)p(di)$ is a lottery on $X$.
Let $\lambda_i$ and $\mu_i$ be families of lotteries on $X$ indexed by $I$. The total preorder $\preceq$ is discretely mixing just in case if $\lambda_i \preceq \mu_i$ for all $i \in I$, then $\int \lambda_i p(di) \preceq \int\mu_ip(di)$ for all discrete probabilities $p$ on $(I, \mathcal I)$.
Question. If $\preceq$ is discretely mixing, then is $\preceq$ totally mixing in the following sense: If $\lambda_i \preceq \mu_i$ for all $i \in I$, then $\int \lambda_i p(di) \preceq \int \mu_i p(di)$ for all probability measures on $(I, \mathcal I)$?
I suspect the answer is negative, but have not found a counterexample.
