I’m confused by the definition of conditional history, because it doesn’t seem to be a generalisation of history. I would expect hF(X|∅)=hF(X), but both of the conditions in the definition of hF(X|∅) are vacuously true if E=∅. This is independent of what H is, so hF(X|∅)=∅. Am I missing something?
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).
I’m confused by the definition of conditional history, because it doesn’t seem to be a generalisation of history. I would expect hF(X|∅)=hF(X), but both of the conditions in the definition of hF(X|∅) are vacuously true if E=∅. This is independent of what H is, so hF(X|∅)=∅. Am I missing something?
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).
When I prove it, I prove and use (a slight notational variation on) these two lemmas.
If hF(X|E)∩hF(Y|E)={}, then hF(X|E)=hF(X|(y∩E)) for all y∈Y.
hF((X∨SY)|E)=hF(X|E)∪⋃x∈XhF(Y|x∩E).
(These are also the two lemmas that I have said elsewhere in the comments look suspiciously like entropy.)
These are not trivial to prove, but they might help.