Real Analysis Exchange Vol. 27(1), 2001/2002, pp. 927–934

Krzysztof Ciesielski^{∗}, Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, USA, email: K Cies@math.wvu.edu, internet:

http://www.math.wvu.edu/~kcies

Janusz Pawlikowski^{†}, Department of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland, email:

pawlikow@math.uni.wroc.pl and

Department of Mathematics, West Virginia University, Morgantown, WV

26506-6310, USA, email: pawlikow@math.wvu.edu

SMALL COMBINATORIAL CARDINAL CHARACTERISTICS AND THEOREMS OF

EGOROV AND BLUMBERG

Abstract

We will show that the following set theoretical assumption

𝔠 = ω_{2}, the dominating number 𝔡 equals to ω_{1}, and there exists an ω_{1}-generated Ramsey ultrafilter on ω

(which is consistent with ZFC) implies that for an arbitrary sequence f_{n}: ℝ → ℝ of uniformly bounded functions there is a set P ⊂ ℝ of cardinality continuum and an infinite W ⊂ ω such that {f_{n} ↾ P : n ∈ W } is a monotone uniformly convergent sequence of uniformly continuous functions. Moreover, if functions f_{n} are measurable or have the Baire property then P can be chosen as a perfect set.

We will also show that cof(N) = ω_{1} implies existence of a magic set and of a function f: ℝ → ℝ such that f ↾ D is discontinuous for every D ∉ N ∩ M.

Our set theoretic terminology is standard and follows that of [8]. In particular, |X| stands for the cardinality of a set X and 𝔠 = |ℝ|. We are using symbols N and M to denote the ideals of Lebesgue measure zero and meager subsets of ℝ, respectively. For the ideal I ∈ {M, N } its cofinality is defined by cof(I) = min{|B|: B ⊂ I generates I}. A set L ⊂ ℝ is a κ-Luzin set if |L| = κ but |L ∩ N| < κ for every nowhere dense subset N of ℝ. Recall that Martin’s Axiom, MA, implies the existence of a 𝔠-Luzin set. The dominating number is defined as

Key Words: cofinality, null sets, uniform convergence, Ramsey ultrafilter, Blumberg theorem, magic set.

Mathematical Reviews subject classification: Primary 26A15, 03E35; Secondary 26A03, 03E17.

Received by the editors November 17, 2000 ∗Papers authored or co-authored by a Contributing Editor are managed by a Managing Editor or one of the other Contributing Editors.

†The work of the second author was partially supported by KBN Grant 2 P03A 031 14.

𝔡 = min {|T |: T ⊂ ω^{ω} & (∀f ∈ ω^{ω})(∃g ∈ T)(∀n < ω) f(n) < g(n)} .

It is well known that ω_{1} ≤ 𝔡 ≤ cof(N). (See e.g. [1].) In this paper we use term Polish space for a complete separable metric space without isolated

points.

1 On a Convergence of Subsequences

This section can be viewed as an extension of the discussion around Egorov’s theorem presented in [12, Ch. 9]. In 1932 Mazurkiewicz [13] proved the following variant of Egorov’s theorem, where a sequence ⟨f_{n}⟩_{n<ω} of real-valued functions is uniformly bounded provided there exists an r ∈ ℝ such that range(f_{n}) ⊂ [−r, r] for every n.

Mazurkiewicz’s Theorem Every uniformly bounded sequence ⟨f_{n}⟩_{n<ω} of real-valued continuous functions defined on a Polish space X has a subsequence which is uniformly convergent on some perfect set P.

Of course Mazurkiewicz’ theorem cannot be proved if we do not assume some regularity of the functions f_{n} even if X = ℝ. But is it at least true that

(∗) for every uniformly bounded sequence ⟨f_{n}: ℝ → ℝ⟩_{n<ω} the conclusion of Mazurkiewicz’ theorem holds for some P ⊂ ℝ of cardinality 𝔠?

The consistency of the negative answer follows from the next example, which is essentially due to Sierpiński [16].^{1} (See [12, pp. 193-194], where it is proved under the assumption of the existence of ω_{1}-Luzin set. The same proof works also for our more general statement.)

Example 1. Assume that there exists a κ-Luzin set. Then for every Polish space X there exists a sequence ⟨f_{n}: X → {0, 1}⟩_{n<ω} with the property that for every W ∈ [ω]^{ω} the subsequence ⟨f_{n}⟩_{n∈W} converges pointwise for less than κ-many points x ∈ X.

In particular, under Martin’s Axiom the above sequence exists for κ = 𝔠.

1Sierpiński constructed this example under the assumption of the Continuum Hypothesis.

Note also that under MA the above example can hold only for κ = 𝔠, since MA implies that

for every set S of cardinality less than 𝔠 every uniformly bounded sequence ⟨f_{n}: S → ℝ⟩_{n<ω} has a pointwise convergent subsequence.

(See [12, p. 195].) Sharper results concerning the above two facts were recently obtained by Fuchino and Plewik [11], in which they relate them to the splitting number 𝔰. (For the definition of 𝔰 see e.g. [1]. For us it is only important that ω_{1} ≤ 𝔰 ≤ 𝔡.) More precisely, the authors show there that: For any X ⊂ [ℝ]^{<𝔰} any sequence ⟨f_{n}: X → [−∞, ∞]⟩_{n<ω} has a subsequence convergent pointwise on X; however for any X ⊂ [ℝ]^{𝔰} there exists a sequence ⟨f_{n}: X → [0, 1]⟩_{n<ω} with no pointwise convergent subsequence.

Our main goal of this section is to prove that (∗) is consistent with (so, by the example, also independent from) the usual axioms of set theory ZFC. To state this precisely we need the following terminology and facts.

A maximal non-principal filter F on ω is said to be Ramsey provided for every B ∈ F and h: [B]^{2} → {0, 1} there exist i < 2 and A ∈ F such that A ⊂ B and h [[A]^{2}] = {i}. We say that a family W ⊂ F generates filter F provided for every F ∈ F there exists a W ∈ W such that W ⊂ F.

Theorem 2. Assume that 𝔡 = ω_{1} and there exists a Ramsey ultrafilter F on ω generated by a family W ⊂ F of cardinality ω_{1}.

Let X be an arbitrary set and ⟨f_{n}: X → ℝ⟩_{n<ω} be a sequence of functions such that the set {f_{n}(x): n < ω} is bounded for every x ∈ X. Then there are sequences: ⟨P_{ξ}: ξ < ω_{1}⟩ of subsets of X and ⟨W_{ξ} ∈ F: ξ < ω_{1}⟩ such that X = ⋃_{ξ<ω_{1}} P_{ξ} and for every ξ < ω_{1}:

the sequence ⟨f_{n} ↾ P_{ξ}⟩_{n∈W_{ξ}} is monotone and uniformly convergent.

The conclusion of Theorem 2 is obvious for sets X with cardinality ≤ ω_{1}, since sets P_{ξ} can be chosen just as singletons. Thus, we will be interested in the theorem only for the sets X of cardinality greater than ω_{1}. If X is a Polish space this leads to 𝔠 = |X| > ω_{1}. Luckily, the assumptions of Theorem 2 are consistent with ZFC+“𝔠 = ω_{2}”. This holds in the iterated perfect set model. More precisely, the fact that in this model we have 𝔠 = ω_{2} and cof(N) = ω_{1} can be found in [1, p. 339]. The fact that in this model there exists a desired Ramsey ultrafilter has been proved in Baumgartner, Laver [2]. (They proved there that there exists a selective ω_{1}-generated ultrafilter on ω. But it is well known that an ultrafilter on ω is selective if and only if it is Ramsey.) All these facts follow also from the axiom CPA, which is a subject of a forthcoming monograph [9]. (Some of the results proved here may also be included in [9] as the examples of interesting consequences of CPA.)

In particular, we get the following corollary which, under additional set theoretical assumptions, generalizes Mazurkiewicz’ theorem and implies (∗).

Corollary 3. It is consistent with ZFC+“𝔠 = ω_{2}” that for each Polish space X and each uniformly bounded sequence ⟨f_{n}: X → ℝ⟩_{n<ω} there exist sequences: ⟨P_{ξ}: ξ < ω_{1}⟩ of subsets of X and ⟨W_{ξ} ∈ [ω]^{ω}: ξ < ω_{1}⟩ such that X = ⋃_{ξ<ω_{1}} P_{ξ} and for every ξ < ω_{1}:

the sequence ⟨f_{n} ↾ P_{ξ}⟩_{n∈W_{ξ}} is monotone and uniformly convergent.

In particular, there exists a ξ < ω_{1} such that |P_{ξ}| = 𝔠.

Moreover, if functions f_{n} are continuous then we can additionally require that all sets P_{ξ} are closed in X.

Proof. The main part follows immediately from the discussion above and the Pigeon Hole Principle. To see the additional part it is enough to note that for continuous functions sets P_{ξ} can be replaced by their closures, since for any sequence ⟨f_{n}: P → ℝ⟩_{n<ω} of continuous functions if ⟨f_{n} ↾ D⟩_{n<ω} is monotone and uniformly convergent for some dense subset D of P then so is ⟨f_{n}⟩_{n<ω}.

Proof of Theorem 2. For every x ∈ X define h_{x}: [ω]^{2} → {0, 1} by putting for every n < m < ω

h_{x}(n, m) = 1 if and only if f_{n}(x) ≤ f_{m}(x).

Since F is Ramsey and W generates F we can find a W_{x} ∈ W and an i_{x} < 2 such that h_{x}[[W_{x}]^{2}] = {i_{x}}. Thus, the sequence S_{x} = ⟨f_{n}(x)⟩_{n∈W_{x}} is monotone. It is increasing when i_{x} = 1 and it is decreasing for i_{x} = 0.

For W ∈ W and i < 2 let P_{W}^{i} = {x ∈ X: W_{x} = W & i_{x} = i}. Then {P_{W}^{i} : W ∈ W & i < 2} is a partition of X and for every W ∈ W and i < 2 the sequence ⟨f_{n} ↾ P_{W}^{i} ⟩_{n∈W} is monotone and pointwise convergent to some function f: P_{W}^{i} → ℝ.

To get uniform convergence note that for every x ∈ P_{W}^{i} there exists an s_{x} ∈ ω^{ω} such that

(∀k < ω) (∀n ∈ W \ s_{x}(k)) |f_{n}(x) − f(x)| < 2^{−k}.

Since 𝔡 = ω_{1}, there exists a T ∈ [ω^{ω}]^{ω_{1}} dominating ω^{ω}. In particular, for every x ∈ P_{W}^{i} there exists a t_{x} ∈ T such that s_{x}(n) ≤ t_{x}(n) for all n < ω. For t ∈ T let

P_{W}^{i} (t) = {x ∈ P_{W}^{i} : t_{x} = t}.

Since every sequence ⟨f_{k} ↾ P_{W}^{i} (t)⟩_{k∈W} is monotone and uniformly convergent, {P_{W}^{i} (t): i < 2, W ∈ W, t ∈ T } is the desired covering, {P_{ξ}: ξ < ω_{1}} of X,

2 cof(N) = ω_{1}, Blumberg Theorem, and Magic Set

In this section we will show two consequences of cof(N) = ω_{1}.

In 1922 Blumberg [4] proved that for every f: ℝ → ℝ there exists a dense subset D of ℝ such that f ↾ D is continuous. This theorem sparked a lot of discussion and generalizations, see e.g. [7, pp. 147–150]. In particular, Shelah [15] showed that there is a model of ZFC in which for every f: ℝ → ℝ there is a nowhere meager subset D of ℝ such that f ↾ D is continuous. The dual measure result, that is the consistency of a statement for every f: ℝ → ℝ there is a subset D of ℝ of positive outer Lebesgue measure such that f ↾ D is continuous, has been also recently established by Rosłanowski and Shelah [14]. Below we note that each of these properties contradicts cof(N) = ω_{1}. (We use here the well known inequality cof(M) ≤ cof(N). See e.g. [1].)

Theorem 4. Let I ∈ {N , M}. If cof(I) = ω_{1} then there exists an f: ℝ → ℝ such that f ↾ D is discontinuous for every D ∈ ¶(ℝ) \ I.

Proof. We will assume that I = N, the proof for I = M being essentially identical.

Let {N_{ξ} ⊂ ℝ^{2} : ξ < ω_{1}} be a family cofinal in the ideal of null subsets of ℝ^{2} and for each ξ < ω_{1} let

N_{ξ}^{∗} = {x ∈ ℝ: (N_{ξ})_{x} ∉ N },

where (N_{ξ})_{x} = {y ∈ ℝ: ⟨x, y⟩ ∈ N_{ξ}}. By Fubini’s theorem each N_{ξ}^{∗} is null. For each x ∈ N_{ξ}^{∗} \ ⋃_{ζ<ξ} N_{ζ}^{∗} we choose f(x) so that

f(x) ∉ ⋃_{ζ<ξ}(N_{ζ})_{x}.

Then function f is as desired.

Indeed, if f ↾ D is continuous for some D ⊂ ℝ then f ↾ D is null in ℝ^{2}. In particular, there exists a ξ < ω_{1} such that f ↾ D ⊂ N_{ξ}. But this means that D ⊂ ⋃_{ζ≤ξ} N_{ζ}^{∗}.

Note that essentially the same proof works if we assume only that cof(I) is equal to the additivity number add(I) of I.

Corollary 5. Assume cof(N) = ω_{1}. Then there exists an f: ℝ → ℝ such that if f ↾ D is continuous then D ∈ N ∩ M.

Proof. Let f_{N} and f_{M} be from Theorem 4 constructed for the ideals N and M, respectively. Let G ⊂ ℝ be a dense G_{δ} of measure zero and put f = [f_{M} ↾ G] ∪ [f_{N} ↾ (ℝ \ G)]. Then this f is as desired.

Recall that a set M ⊂ ℝ is a magic set (or set of range uniqueness) if for every different nowhere constant functions f, g ∈ C(ℝ) we have f[M] ≠ g[M]. It has been proved by Berarducci and Dikranjan [3, thm. 8.5] that a magic set exists under CH. We like to note here that the same is implied by a much weaker assumption that cof(M) = ω_{1}. However, the existence of a magic set is independent of ZFC, as proved by Ciesielski and Shelah in [10].

Proposition 6. If cof(M) = ω_{1} then there exists a magic set.

Proof. An uncountable set L ⊂ ℝ is a 2-Luzin set provided for every disjoint subsets {x_{ξ}: ξ < ω_{1}} and {y_{ξ}: ξ < ω_{1}} of L, where the enumerations are oneto-one, the set of pairs {⟨x_{ξ}, y_{ξ}⟩: ξ < ω_{1}} is not a meager subset of ℝ^{2}. In [5, prop. 4.8] it was noticed that every ω_{1}-dense 2-Luzin set is a magic set. It is also a standard and easy diagonal argument that cof(M) = ω_{1} implies the existence of a ω_{1}-dense 2-Luzin set. (The proof presented in [17, prop. 6.0] works also under the assumption cof(M) = ω_{1}.) So, cof(M) = ω_{1} implies that there is a magic set.

Recall also that the existence of a magic set for the class D^{1} of all differentiable functions can be proved in ZFC. This follows from [6, thm. 3.1], since every function from D^{1} belongs to the class (T_{2}). (Compare also [6, cor. 3.3 and 3.4].)

References

[1] T. Bartoszyński, H. Judah, Set Theory, A K Peters, 1995.

[2] J. Baumgartner, R. Laver, Iterated perfect-set forcing, Ann. Math. Logic 17 (1979), 271–288.

[3] A. Berarducci, D. Dikranjan, Uniformly approachable functions and UA spaces, Rend. Ist. Matematico Univ. di Trieste 25 (1993), 23–56.

[4] H. Blumberg, New properties of all real functions, Trans. Amer. Math. Soc. 24 (1922), 113–128.

[5] M. R. Burke, K. Ciesielski, Sets on which measurable functions are determined by their range, Canad. J. Math. 49 (1997), 1089–1116. (Preprint^{⋆} available.^{2})

[6] M. R. Burke, K. Ciesielski, Sets of range uniqueness for classes of continuous functions, Proc. Amer. Math. Soc. 127 (1999), 3295–3304. (Preprint^{⋆} available.)

2Preprints marked by ^{⋆} are available in electronic form from Set Theoretic Analysis Web Page: http://www.math.wvu.edu/˜ kcies/STA/STA.html

[7] K. Ciesielski, Set theoretic real analysis, J. Appl. Anal. 3(2) (1997), 143– 190. (Preprint^{⋆} available.)

[8] K. Ciesielski, Set Theory for the Working Mathematician, London Math. Soc. Stud. Texts 39, Cambridge Univ. Press 1997.

[9] K. Ciesielski, J. Pawlikowski, Covering property axiom CPA, work in progress^{⋆}.

[10] K. Ciesielski, S. Shelah, Model with no magic set, J. Symbolic Logic 64(4) (1999), 1467–1490. (Preprint^{⋆} available.)

[11] S. Fuchino, Sz. Plewik, On a theorem of E. Helly, Proc. Amer. Math. Soc. 127(2) (1999), 491–497.

[12] A. B. Kharazishvili, Strange Functions in Real Analysis, Pure and Applied Mathematics 229, Marcel Dekker, 2000.

[13] S. Mazurkiewicz, Sur les suites de fonctions continues, Fund. Math. 18 (1932), 114–117.

[14] A. Rosłanowski, S. Shelah, Measured Creatures, preprint.

[15] S. Shelah, Possibly every real function is continuous on a non–meagre set, Publ. Inst. Mat. (Beograd) (N.S.) 57(71) (1995), 47–60.

[16] W. Sierpiński, Remarque sur les suites infinies de fonctions (Solution d’un problème de M. S. Saks), Fund. Math. 18 (1932), 110–113.

[17] S. Todorcevic, Partition problems in topology, Contemp. Math. 84, Amer. Math. Soc. 1989.

934