In most introductory courses, a martingale $Y$ is defined as a stochastic process
$$Y: T \times \Omega \to S$$
,which satisfies certain conditions. ($\Omega$ is a probability space and a filtration on it is assumed to be given.)
The index set $T$ is then defined to be a totally ordered subset of $\mathbb R$, e.g. $T = \mathbb N$ or $T = [0,1]$.
Now, I am interested in the case where $T = [0,1]^2$. In this case $T$ is not completely ordered, but a directed order still exists. Do any good resources (preferably beginner friendly, i.e. assuming only knowledge of the simpler martingales defined above) exist which define martingales and derive their properties for such "index" sets?
I am particularly interested in the Optional Stopping theorem in the above scenario.