I remember my lecturer saying that in some cases there will be no other null set than the trivial one (the empty set), but I can't remember exactly the condition. I've been thinking and convinced my self that for finite sets, equipped with the power set and a probability measure defined through the counting measure the statement would be true pretty obviously, but how about a countable state space?
My reasoning is that it's a big deal in conditional expectations, since we would then not have to work with versions of the conditional expectation.
Hope someone can help give me some insight,
Henrik