18. Large cardinals

EXERCISES
E18.1. Let κ be an uncountable regular cardinal. We define S < T iff S and T are
stationary subsets of κ and the following two conditions hold:
(1) {α ∈ T : cf(α) ≤ ω} is nonstationary in κ.
(2) {α ∈ T : S ∩ α is nonstationary in α)} is nonstationary in κ.
Prove that if ω < λ < µ < κ, all these cardinals regular, then E κ λ < Eκ µ , where
and similarly for E κ µ .
E κ λ = {α < κ : cf(α) = λ},
E18.2. Continuing exercise E18.1: Assume that κ is uncountable and regular. Show that
the relation < is transitive.
E18.3. If κ is an uncountable regular cardinal and S is a stationary subset of κ, we define
Tr(S) = {α < κ : cf(α) > ω and S ∩ α is stationary in α}.
Suppose that A, B are stationary subsets of an uncountable regular cardinal κ and A < B.
Show that Tr(A) is stationary.
E18.4. (Real-valued measurable cardinals) We describe a special kind of measure. A
measure on a set S is a function µ : P(S) → [0, ∞) satisfying the following conditions:
(1) µ(∅) = 0 and µ(S) = 1.
(2) If µ({s}) = 0 for all s ∈ S,
(3) If 〈X i : i ∈ ω〉 is a system of pairwise disjoint subsets of S, then µ( ⋃ i∈ω X i) =
∑
i∈ω µ(X i). (The X i ’s are not necessarily nonempty.)
Let κ be an infinite cardinal. Then µ is κ-additive iff for every system 〈X α : α < γ〉 of
nonempty pairwise disjoint sets, with γ < κ, we have
( ) ⋃
X α
α