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).
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.