Interchangeability between knowing an event obtains with probability 1 and knowing an event obtains with absolute certainty? In the literature of interactive epistemology, for a player, knowing an event obtains with probability one and knowing an event obtains with absolute certainty are different. Is there a nontrivial (say, it could be the case, an event is a proper subset of the event that player $i$ knows this event obtains with probability one, but never the case for knowledge operator with absolute certainty) proposition that could serve as an example to show they're not interchangeable in the language which consists of two corresponding knowledge operators, a set of states of world, set relations, set operations, and all logical connectives.
If you take the standard Aumann model, but allow for states occuring with probability zero, you can taje some nonempty event $N$ such that for the agents prior $p$, one has $p(N)=0$. Let $K\cap N=\emptyset$ and $p(K)&gt;0$. If the agent knows $K\cup N$ but not $K$, then she assigns probability $1$ to $K$, even though she does not know $K$.