Let $M$ be a $n$-dimensional stochastic process. Suppose that $\forall i \leq n$, there exists a measure $\mathbb{P}^{(i)}$ such that the component $M^{(i)}$ is a $\mathbb{P}^{(i)}$-martingale. Is it true that $M$ is a $\otimes_{i = 1}^n \mathbb{P}^{(i)}$-martingale ?
I would tend to say yes, as a multivariate process is a martingale if and only if each component of this process is a martingale, but I am not completely sure as I consider a different probability measure for each component.
Thank you in advance for your help.