18. Large cardinals

{X ⊆ A : µ(X) = µ(A)} is a κ-complete nonprincipal ultrafilter on A. Conclude that κ is
a measurable cardinal if there exist such µ and A.
E18.6. Prove that if κ is real-valued measurable then either κ is measurable or κ ≤ 2 ω .
Hint: if there do not exist any µ-atoms, construct a binary tree of height at most ω 1 .
E18.7. Let κ be a regular uncountable cardinal. Show that the diagonal intersection of
the system 〈(α + 1, κ) : α < κ〉 is the set of all limit ordinals less than κ.
E18.8. Let F be a filter on a regular uncountable cardinal κ. We say that F is normal
iff it is closed under diagonal intersections. Suppose that F is normal, and (α, κ) ∈ F for
every α < κ. Show that every club of κ is in F. Hint: use exercise E18.7.
E18.9. Let F be a proper filter on a regular uncountable cardinal κ. Show that the
following conditions are equivalent.
(i) F is normal
(ii) For any S 0 ⊆ κ, if κ\S 0 /∈ F and f is a regressive function defined on S 0 , then
there is an S ⊆ S 0 with κ\S /∈ F and f is constant on S.
E18.10. A probability measure on a set S is a real-valued function µ with domain P(S)
having the following properties:
(i) µ(∅) = 0 and µ(S) = 1.
(ii) If X ⊆ Y , then µ(X) ≤ µ(Y ).
(iii) µ({a}) = 0 for all a ∈ S.
(iv) If 〈X n : n ∈ ω〉 is a system of pairwise disjoint sets, then µ( ⋃ n∈ω X n) =
∑
n∈ω µ(X n). (Some of the sets X n might be empty.)
Prove that there does not exist a probability measure on ω 1 . Hint: consider an Ulam
matrix.
E18.11. Show that if κ is a measurable cardinal, then there is a normal κ-complete nonprincipal
ultrafilter on κ. Hint: Let D be a κ-complete nonprincipal ultrafilter on κ. Define
f ≡ g iff f, g ∈ κ κ and {α < κ : f(α) = g(α)} ∈ D. Show that ≡ is an equivalence relation
on κ κ. Show that there is a relation ≺ on the collection of all ≡-classes such that for all
f, g ∈ κ κ, [f] ≺ [g] iff {α < κ : f(α) < g(α)} ∈ D. Here for any function h ∈ κ κ we use [h]
for the equivalence class of h under ≡. Show that ≺ makes the collection of all equivalence
classes into a well-order. Show that there is a ≺ smallest equivalence class x such that
∀f ∈ x∀γ < κ[{α < κ : γ < f(α)} ∈ D. Let E = {X ⊆ κ : f −1 [X] ∈ D}. Show that E
satisfies the requirements of the exercise.
Reference
Kanamori, A. The higher infinite. Springer 2005, 536pp.
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