SEQUENTIAL G_{δ}–SETS AND MEASURES

Roger H. Marty

Cleveland State University

For any set Y we equip the power set P (Y) with a compact Hausdorff topology by taking as subbasic open sets all sets of the forms {D ⊆ Y | y ∈ D} and {D ⊆ Y | y ∉ D} where y varies throughout Y. For any class X of sets, the union of this class is a set ∪X and its power set P (∪X) may be equipped with a topology as above. Considered as a subset of P (∪X), X might be both sequentially closed in P (∪X) and a sequential G_{δ}–set (the intersection of countably many sequentially open sets) in P (∪X). If this is the case, we say that X is a Mazur class.

For any set Y, we say that Y is Mazur reducible provided every Mazur class of subsets of Y that contains all finite subsets of Y must also contain Y itself. A cardinal number m is said to be Mazur reducible if there is a Mazur reducible set of cardinality m. (If a set is Mazur reducible, then so are all sets of the same cardinality as this set.)

Mazur [2] proved that all cardinal numbers less than the smallest inaccessible cardinal number are Mazur reducible. Ulam [3] proved that each nonmeasurable cardinal number is non–realmeasurable provided 2^{ω} is non–realmeasurable.

For any set Y, we say that Y is modified Mazur reducible provided every Mazur class of subsets of Y that is finitely additive and contains all finite subsets of Y must also contain Y itself. As above, we say that a cardinal number m is modified Mazur reducible if there is a modified Mazur reducible set of cardinality m.

It follows from [1, Theorem 1] that all Mazur reducible cardinals are modified Mazur reducible.

\underline{Theorem}. Every nonmeasurable cardinal number is modified Mazur reducible if and only if 2^{ω} is modified Mazur reducible.

\underline{Proof}. Since 2^{ω} is nonmeasurable, it remains to prove that if 2^{ω} is modified Mazur reducible, then m is modified Mazur reducible for every nonmeasurable cardinal number m.

\underline{Lemma 1}. Let Y denote a set of nonmeasurable cardinality. If X is a Mazur class of subsets of Y that is finitely additive and contains all finite subsets of Y, then for every B ∈ P (Y) − X there is a partition of B into two (disjoint) sets both belonging to P (Y) − X or both belonging to X.

\underline{Proof}. Suppose the conclusion is false; i.e., there is a set B ∈ P (Y) − X such that for every partition S/T of B exactly one of S and T belongs to X. But this defines a measure on B and hence, is inconsistent with the fact that Y has nonmeasurable cardinality.

\underline{Lemma 2}. Suppose Y is any set, and X is a Mazur class of subsets of Y that is finitely additive and contains all finite subsets of Y, and for every B ∈ P (Y)−X there is a partition of B into two sets both belonging to P (Y) − X or both belonging to X. There there is a family {A_{ξ} | ξ ∈ Ξ} of mutually disjoint members of X such that card (Ξ) ≤ 2^{ω} and Y = ∪{A_{ξ} | ξ ∈ Ξ}.

\underline{Proof}. Suppose the conclusion is false; i.e., every representation of Y as a union of mutually disjoint members of X must be formed by a family of cardinality greater than 2^{ω}. By hypothesis, the set Y partitions into B_{1}^{0}, B_{2}^{0} ∈ P (Y) − X. Take S_{0} = ∅. Again B_{1}^{0} and B_{2}^{0} each partition into sets B′_{i} ∈ P (Y) − X (1 ≤ i ≤ 4) and we take S_{1} = ∅ again.

Assume that S_{ξ} ⊆ X and B_{n}^{ξ} ∈ P (Y) − X have been defined for every ξ < η < ω_{1} and satisfy the following conditions:

(1) card S_{ξ} ≤ 2^{ω}, and the sets in S_{ξ} are mutually disjoint;

(2) S_{ξ′} ⊆ S_{ξ} for ξ′ < ξ;

(3) B_{n}^{ξ} ∩ B_{m}^{ξ′} = ∅, or B_{n}^{ξ} ⊆ B_{m}^{ξ′} and B_{m}^{ξ′} − B_{n}^{ξ} ∈ P (Y) − X for ξ′ < ξ; and

(4) (∪S_{ξ}) ∪ (∪_{n}B_{n}^{ξ}) = Y, (∪S_{ξ}) ∩ (∪_{n}B_{n}^{ξ}) = ∅.

Consider C = {H ∈ P (Y) − {∅} | there are n_{ξ} with H = ∩{B_{n_{ξ}}^{ξ} | ξ < η}}. By assumption and condition (1), Y ≠ ∪S_{ξ} for every ξ < η. Let p ∈ Y − ∪S_{ξ} for every ξ < η. Thus, for every ξ < η there is a set B_{n_{ξ}}^{ξ} containing p. Thus, C ≠ ∅. The sets in C are mutually disjoint since their defining sequences must differ for some ξ < η.

Let S_{η} = (∪{S_{ξ} | ξ < η} ∪ {H ∈ C | H ∈ X}. Then Y = (∪S_{η}) ∪ (∪(C − S_{η})) and C − S_{η} is a countable family of sets in P (Y) − X because P (Y) − X is a countable union of sequentially closed sets, X contains all finite subsets of Y, the sets in C are mutually disjoint, and the pigeon–hole principle holds. Since card (S_{η}) ≤ 2^{ω}, C − S_{η} ≠ ∅. Applying the hypothesis to each set in C − S_{η} yields a countable sequence B_{1}^{η}, B_{2}^{η}, · · ·. It is routine to verify that conditions (1)–(4) are satisfied.

Since Y ≠ ∪{∪{S_{ξ} | ξ < ω_{1}}}, there is a p ∈ Y − ∪{∪{S_{ξ} | ξ < ω_{1}}}. Thus, there is an n_{ξ} for every ξ < ω_{1}, for which p ∈ ∩{B_{n_{ξ}}^{ξ} | ξ < ω_{1}}. Then {B_{n_{ξ}}^{ξ} − B_{n_{ξ+1}}^{ξ+1} | ξ < ω_{1}} is an uncountable, mutually disjoint collection of elements of P (Y) − X.

Because P (Y) − X is a countable union of sequentially closed sets, one of these sets, say X_{0}, must therefore contain uncountably many elements of the above mutually disjoint collection, and any sequence, with distinct terms, of these sets converges to the empty set. This is a contradiction since the empty set belongs to X, X_{0} is sequentially closed, and X is disjoint from X_{0}.

To complete the proof of the theorem, we suppose Y is a set of nonmeasurable cardinality and that X is a Mazur class of subsets of Y that is finitely additive and contains all finite subsets of Y. By Lemmas 1 and 2, there is a family {A_{ξ} | ξ ∈ Ξ} of mutually disjoint members of X such that Y = ∪{A_{ξ} | ξ ∈ Ξ} and card (Ξ) ≤ 2^{ω}.

Define a map ϕ from Y onto Ξ by ϕ(y) = ξ for y ∈ A_{ξ}. It is easy to check that X_{ϕ} = {Γ ⊆ Ξ | ϕ^{−1}[Γ] ∈ X} is a Mazur class of subsets of Ξ that is finitely additive and contains all finite subsets of Ξ.

By hypothesis, 2^{ω} is modified Mazur reducible (and hence, all cardinal numbers less than 2^{ω} are also) so that Ξ ∈ X_{ϕ}. Thus, Y = ϕ^{−1}[Ξ] ∈ X, which was to be proved.

\underline{References}

1. R. H. Marty, “Mazur’s Theorem and Banach Measures,” Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 25 (1977), 455–456.

2. S. Mazur, “On Continuous Mappings on Cartesian Products,” Fund. Math., 39 (1952), 229–238.

3. S. Ulam, “Zur Masstheorie in der Allgemeinen Mengenlehre,” Fund. Math., 16 (1930), 140–150. æ