A lower density operator for the Borel algebra
aa r X i v : . [ m a t h . GN ] F e b A LOWER DENSITY OPERATOR FOR THE BOREL ALGEBRA
MAREK BALCERZAK AND SZYMON G LA¸ B
Abstract.
We prove that the existence of a Borel lower density operator (a Borel lifting) with respectto the σ -ideal of countable sets, for an uncountable Polish space, is equivalent to the ContinuumHypothesis. Let S be a σ -algebra of subsets of a nonempty set X and let J ⊆ S be a σ -ideal. We write A ∼ B whenever A △ B ∈ J . A mapping Φ : S → S is called a lower density operator (respectively, a lifting )with respect to J if it satisfies the following conditions (1)–(4) (respectively, (1)–(5)):(1) Φ( X ) = X and Φ( ∅ ) = ∅ ,(2) A ∼ B = ⇒ Φ( A ) = Φ( B ) for every A, B ∈ S ,(3) A ∼ Φ( A ) for every A ∈ S ,(4) Φ( A ∩ B ) = Φ( A ) ∩ Φ( B ) for every A, B ∈ S ,(5) Φ( A ∪ B ) = Φ( A ) ∪ Φ( B ) for every A, B ∈ S .The problem of the existence of liftings together with their various applications were widely dis-cussed in the monograph [3] and in the later survey [9]. If S is the σ -algebra of Borel sets in a givenHausdorff space, then the respective operator Φ satisfying conditions (1)–(5) is called a Borel lifting.Note that von Neumann and Stone [6] proved the existence of a lifting for a Borel measure space on[0 ,
1] under the assumption of the continuum hypothesis ( CH ). A simple proof of the same result wasthen given by Musia l [5]. This was later generalized by Mokobodzki [4] and Fremlin [1] who showedthat, subject to CH , any σ -finite measure space with the measure algebra of cardinality ≤ ω has alifting. On the other hand, Shelah [7] proved that it is consistent with ZFC that there exists no Borellifting for Lebesgue measure on [0 , S is the σ -algebra B of Borel subsets of anuncountable Polish space X and J is the σ -ideal [ X ] ≤ ω of all countable subsets of X . Since any twouncountable Borel subsets of Polish spaces are Borel isomorphic [8, Thm 3.3.13], it does not matterwhich Polish space is considered. Theorem 1.
For an uncountable Polish space X , the following conditions are equivalent: (i) CH ; (ii) there exists a lifting Φ :
B → B with respect to [ X ] ≤ ω ; (iii) there exists a lower density operator Φ :
B → B with respect to [ X ] ≤ ω .Proof. Implication (i) = ⇒ (ii) follows from [5, Thm 1]. Implication (ii) = ⇒ (iii) is obvious.To prove (iii) = ⇒ (i) assume ¬ CH . We work with X := R × R . Enumerate R as { x α : α < c } .Suppose that Φ : B → B is a lower density operator with respect to [ X ] ≤ ω . Let Q α := Φ( P α ) where P α := { x α } × R for α < c . Note that if α = β then Q α ∩ Q β = Φ( P α ∩ P β ) = ∅ by (4) and (1). Let π : R × R → R be given by π ( x, y ) := y . Claim.
There is x ∈ R such that { α < c : x ∈ π [ Q α ] } is uncountable. Proof of Claim.
Suppose that |{ α < c : x ∈ π [ Q α ] }| ≤ ω for each x ∈ R . Let L α := { β < c : x α ∈ π [ Q β ] } for α < c . Then | S α<ω L α | ≤ ω by our supposition. By ¬ CH , the set c \ S α<ω L α is nonempty (of cardinality c ). Moreover, { x α : α < ω } ∩ π [ Q ξ ] = ∅ for each ξ ∈ c \ S α<ω L α .Thus { x α : α < ω } ⊆ R \ π [ Q ξ ] = π [ P ξ ] \ π [ Q ξ ] ⊆ π [ P ξ \ Q ξ ] which gives a contradiction since | π [ P ξ \ Q ξ ] | ≤ | P ξ \ Φ( P ξ ) | ≤ ω by (3).Take x ∈ R as in the Claim. Consider the closed set P := R × { x } . Then | P ∩ P α | = 1 andΦ( P ) ∩ Q α = Φ( P ) ∩ Φ( P α ) = Φ( P ∩ P α ) = ∅ for each α < c , by (4), (2) and (1). Therefore Mathematics Subject Classification.
Primary: 28A51; Secondary: 03E50, 03E35.
Key words and phrases.
Borel lifting, lower density operator, Continuum Hypothesis. Φ( P ) ∩ S α< c Q α = ∅ . On the other hand, | P ∩ S α< c Q α | > ω by the choice of x . Thus P \ Φ( P ) isuncountable and we reach a contradiction with (3). (cid:3) Note that implication (iii) = ⇒ (ii) follows from [2, Theorem 2.8].The above theorem answers a question posed by Jacek Hejduk during his invited talk given onthe Conference on Real Function Theory in Star´a Lesn´a in September 2016. He asked about theexistence of a lower density operator on B ( R ) with respect to [ R ] ≤ ω . Let us mention that lowerdensity operators play an important role in constructions of density like topologies; see [10], [2]. Acknowledgement.
The first author would like to thank Kazimierz Musia l and Jacek Hejdukfor their useful comments.
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Institute of Mathematics, Lodz University of Technology, W´olcza´nska 215, 93-005 L´od´z, Poland
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