Supplement to Common Knowledge
Proof of Proposition 2.4
Proposition 2.4.If ω ∈ K*N(E) and E ⊆ F, then ω ∈ K*N(F).
Proof.
If E ⊆ F, then as we observed earlier,
Ki(E) ⊆
Ki(F), so
K1N(E) =
∩
i∈NKi(E) =
∩
i∈NKi(F) = K1N(F)
If we now set E′ = KnN(E) and F′ = KnN(F), then by the argument just given we have
Kn+1N(E) = K1N(E′) ⊆ K1N(F′) = Kn+1N(F)
so we have mth level mutual knowledge for every n ≥ 1.
| Hence if ω ∈ |
∞ ∩ n=1 |
KnN(E) then ω ∈ |
∞ ∩ n=1 |
KnN(F). □ |
