On the point about real-life measurements: we can observe events of probability 0, such as 77.3±0.1 when the distribution was uniform on [0,1]. What we can’t observe are events that are non-open sets. I actually think that “finitely observable event” is a great intuitive semantics for the topological concept of “open set”; see Escardó′s Synthetic Topology.
My proposal (that a probabilistic theory can be falsified when an observed event is disjoint from its support) is equivalent to saying that a theory can be falsified by an observation which is a null set, provided we assume that any event we could possibly observe is necessarily an open set (and I think we should indeed set up our topologies so that this is the case).
On the point about real-life measurements: we can observe events of probability 0, such as 77.3±0.1 when the distribution was uniform on [0,1]. What we can’t observe are events that are non-open sets. I actually think that “finitely observable event” is a great intuitive semantics for the topological concept of “open set”; see Escardó′s Synthetic Topology.
My proposal (that a probabilistic theory can be falsified when an observed event is disjoint from its support) is equivalent to saying that a theory can be falsified by an observation which is a null set, provided we assume that any event we could possibly observe is necessarily an open set (and I think we should indeed set up our topologies so that this is the case).