In fact, the familiar tools of measure theory can take this intuition further. If a random variable is measurable with respect to a sigma algebra, then knowing which element of that sigma algebra my state is in actually is sufficient to pinpoint the value of a random variable.
Maybe to make this more concrete:
Let's say I'm going to do two coinflips. My probability space is {HH, HT, TH, TT}. You can check for yourself that the sigma algebra generated by {{HH, HT}}, {TT, TH}} is not the trivial one- this is the sigma algebra that represents "Knowing the value of the first flip, but not the second".
If we let X_first and X_second be 1 or 0 if the first or second flip is H or T respectively, then X_first is measurable with respect to this sigma algebra, but X_second is not.
With Martingales and other stochastic processes, we don't generally have just one sigma algebra, but a sequence of sigma algebras called a "filtration", where each sigma algebra is finer than the last (ie, contains more sets, therefore gives you more measurable random variables). This filtration sort of defines the stochastic process- it's encoding the slow drip of extra information as the stochastic process evolves over time.
I haven't yet figured out what the author is doing with the sigma algebra machinery. I had thought that the event set of a probability space was intuitively the power set of the sample set, and that the machinery of sigma algebras was only added to avoid pathologies like non-measurable sets (think Banach-Tarski paradox). Maybe there is more to it?
I would like to understand this better and I do plan to read the article, but I had found the Wikipedia article on martingales to be reasonably understandable a while back. The good thing about this article is it shows applications.