I am trying to understand what it means for some event $A$ to occur almost surely. I understand that the definition is that if $(\Omega, \mathcal{F}, P)$ is a probability triple, then $A \in \mathcal{F}$ happens almost surely if $P(A) = 1$.
My understanding here is that this differs from occurring surely as there may exist events in which $\omega \notin A$, but that they happen with probability zero.
However, when I think of the example where we have a single roll of a die, the sample space being $\Omega = \{1,2,3,4,5,6\}$, it appears that $P(\omega\in \Omega) = 1$. However, this seems to only imply that this happens almost surely, that there are events which could occur, but happen with probability zero. I am not clear what those "null set" events are. Could someone tell me what those might be?