3y1

I'm confused by the definition of conditional history, because it doesn't seem to be a generalisation of history. I would expect , but both of the conditions in the definition of are vacuously true if . This is independent of what is, so . Am I missing something?

33y

E is the event you are conditioning on, so the thing you should expect is that hF(X|S)=hF(X), which does indeed hold.

Thanks, that makes sense! Could you say a little about why the weak union axiom holds? I've been struggling to prove that from your definitions. I was hoping that hF(X|z,w)⊆hF(X|z) would hold, but I don't think that hF(X|z) satisfies the second condition in the definition of conditional history for hF(X|z,w).