I've been studying martingale property under change of measure and I came up with a following observation.
I took a random example with a state space $\Omega = \{ -2,1,2\}$ and two equivalent probability measures $P = \{ 2/5, 2/5, 1/5\}$ and $Q = \{ 3/7, 2/7, 2/7\}$. A stochastic process $Z_{n} = \sum_{i=1}^{n} X_{i}$ is then a martingale with respect to both $P$ and $Q$ where $X_{i}: \Omega \rightarrow \mathbb{R}$ is a random variable.
Then it seems true that $Z_{n}$ remains a martingale for any convex mixture of probability measures $P$ and $Q$. Is this a general property? Is it true that whenever a stochastic process is a martingale for a number of probability measures, it remains so for any (countable?) convex mixture of these? What kind of (implicit) hypothesis I should be aware of?