Combinatorial Variants of Lebesgue s Density Theorem

Transcription

1 Combinatorial Variants of Lebesgue s Density Theorem Philipp Schlicht joint with David Schrittesser, Sandra Uhlenbrock and Thilo Weinert September 6, 2017 Philipp Schlicht Variants of Lebesgue s Density Theorem 1/19

2 Lebesgue s Density Theorem Suppose that (X, d, µ) is a Polish metric space with a Borel measure µ. An element x of X is a µ-density point of a subset A of X if Theorem lim inf ɛ>0 µ(b ɛ (x) A)) µ(b ɛ (x)) = 1. (1) (Lebesgue) Suppose that A is a Lebesgue measurable subset of R n with the Lebesgue measure and D L (A) is the set of Lebesgue density points of A. Then µ(a D L (A)) = 0. (2) (Miller) Suppose that (X, d, µ) is an ultrametric Polish space with a finite Borel measure µ and A X is µ-measurable. Let D L,µ (A) be the set of µ-density points of A. Then µ(a D L,µ (A)) = 0. Philipp Schlicht Variants of Lebesgue s Density Theorem 2/19

3 Lebesgue s Density Theorem Theorem (Vitali s Covering Theorem) Given a collection of open balls centered at the points of a set A R n that contains arbitrary small balls at each point in A, there is a disjoint subcollection that covers A except for a null set. Proof of Lebesgue s Density Theorem. We claim that A ɛ = {x R n µ(b lim sup r(x)\a) r 0 µ(b r(x)) for all ɛ > 0. > ɛ} is a null set For any δ > 0, let U δ be an open set with A ɛ U δ and µ(u δ \ A ɛ) < δ. We obtain a collection C from Vitali s Covering Theorem for the collection of all open balls B U δ with µ(b\a) > ɛ at elements of A µ(b) ɛ. Then ɛµ(a ɛ) ɛ B C µ(b) < B C µ(b \ A) µ(u δ \ A) < δ. Since this holds for all δ > 0, we have µ(a ɛ) = 0. Philipp Schlicht Variants of Lebesgue s Density Theorem 3/19

4 Counterexamples If (X, d) is a metric space, d is called doubling if for some n N, any open ball B 2r (x) can be covered by n balls of radius r. Theorem (Käenmäki-Rajala-Suomala) There is a finite Borel measure ν and a complete doubling metric δ on the Cantor space, compatible with the standard topology, such that some closed set C of positive measure has no ν-density points. Theorem (Andretta-Costantini-Camerlo) For any Polish measure space (X, d, µ) there is a compatible metric δ such that (X, δ, µ) does not satisfy Lebesgue s density theorem. Philipp Schlicht Variants of Lebesgue s Density Theorem 4/19

5 Question Can the Lebesgue density theorem be generalized to other ideals instead of the ideal of null sets, in particular the σ-ideals defined by tree forcings? Philipp Schlicht Variants of Lebesgue s Density Theorem 5/19

6 Tree forcings and their ideals = 2 or = ω. We say P is a tree forcing iff the conditions in P are perfect subtrees of <ω ordered by inclusion such that for all T P and all s T we have that {t T s t or t s} P. (Ikegami) Suppose that P is a tree forcing and A is a subset of ω. (i) A N P if for every T P, there is some S P with S T and [S] A =. A set A in N P is also called P-null. (ii) I P is the σ-ideal generated by N P. (iii) A IP if for every T P, there is some S P with S T and [S] A I P. Philipp Schlicht Variants of Lebesgue s Density Theorem 6/19

7 Example: Random forcing Let µ denote the uniform measure on ω 2. Random forcing B is the tree forcing consisting of perfect trees T 2 <ω such that µ([t ]) > 0 and for all s T, µ([{t T s t or t s}]) > 0. A subset A of ω 2 is B-measurable iff for every T B there is some S B with S T such that either [S] A I B or [S] A c I B. What is a B-density point? Philipp Schlicht Variants of Lebesgue s Density Theorem 7/19

8 Density points for Random forcing (i) Suppose that A is a B-measurable subset of ω 2. Suppose that x ω 2. Then x is a B-translation density point of A if for every T B and s = stem T, there is some n 0 such that for all n n 0, f x n f 1 s [T ] A / I B. (ii) Let D B tr(a) denote the set of translation density points of A. (iii) We say that B has the translation density property if for every B-measurable subset A of ω 2, A D B tr(a) I B. Here f s (x) = s x and f 1 s [T ] = [T/s] = {t s t [T ]}. Philipp Schlicht Variants of Lebesgue s Density Theorem 8/19

9 Density points for Random forcing Lemma Suppose that A is a B-measurable subset of ω 2. (a) If lim inf n µ n (x, A) = 1, then x is a B-translation density point of A. (b) If lim inf n µ n (x, A) = 0, then x is not a B-translation density point of A. (c) If lim inf n µ n (x, A) (0, 1), then x can be but does not have to be a B-translation density point of A. Corollary For every B-measurable set A, D L (A) = I B D B tr(a). In particular B has the translation density property. Philipp Schlicht Variants of Lebesgue s Density Theorem 9/19

10 Density points for ideals Let I always denote a σ-ideal on the Borel subsets of ω. (a) A map g : A B between Borel sets A, B is I-invariant if for every Borel set X B, we have X I if and only if g 1 [X] I. (b) Let Bor(I) denote the set of all I-invariant Borel isomorphisms g : ω ω. (c) We say B is I-positive iff B / I. Let I + denote the I-positive sets. Philipp Schlicht Variants of Lebesgue s Density Theorem 10/19

11 Density points for ideals Suppose that A is a subset of ω and Γ is a subgroup of Bor(I). (i) An element x of ω is an I-density point of A w.r.t. Γ if for every I-positive Borel set B, there is some s <ω and some n B such that for all n n B and all g Γ, (f x n g f 1 s )[B] A / I. Let D I,Γ (A) denote the set of I-density points of A w.r.t. Γ. (ii) An element x of ω is a strong I-density point of A if there is some n 0 such that for all n n 0, we have f 1 x n [Ac N x n ] I. Let D I,strong (A) denote the set of strong I-density points of A. Note that D I,strong (A) D I,Bor(I) (A) D I,Γ (A) D I,{id} (A). Philipp Schlicht Variants of Lebesgue s Density Theorem 11/19

12 Density points We say that I has the density property w.r.t. Γ if for all Borel subsets A of ω, A D I,Γ (A) I. Philipp Schlicht Variants of Lebesgue s Density Theorem 12/19

13 Density points (i) P is homogeneous if for all S, T P, there is some U T and f : [S] [U] in Bor(I P ). (ii) P is nondegenerate if it is not equivalent to Cohen forcing and s S P s stem S. (iii) (Friedman-Khomskii-Kulikov) A tree forcing P is topological if for all S, T P with [S] [T ], there is U P with [U] [S] [T ]. For a topological tree forcing P, we let τ P be the topology on ω with basis {[T ] T P}. Lemma Assume that P is a homogeneous, nondegenerate, topological tree forcing and let I = IP. Then for all I-measurable A, D I,strong (A) = D I,Bor(I) (A). Philipp Schlicht Variants of Lebesgue s Density Theorem 13/19

14 Density property Suppose that P is a topological tree forcing and let A be a I P -measurable subset of ω which has the property of Baire in τ P. An x ω is a P-topological density point of A if x U = D P top(a), where U is the unique open subset in τ P such that A U I P. Theorem Suppose that P is a homogeneous, nondegenerate, topological tree forcing with the ccc w.r.t. I = IP. Suppose that for every T P there is an S T such that every x [S] is an I-density point of [T ] w.r.t. Bor(I). Then for every I-measurable A, D I,Bor(I) (A) = D I,strong (A) = I D P top(a). Therefore I has the density property w.r.t. Bor(I). Philipp Schlicht Variants of Lebesgue s Density Theorem 14/19

15 Stem-linked forcings Suppose that P is a tree forcing on <ω. Then P is stem-linked if for all S, T P, if stem S stem T and stem T S, then S and T are compatible. Stem-linked implies σ-linked, ccc w.r.t. IP, and topological. Lemma Let P be a stem-linked tree forcing on <ω. Then for every T P we have that every x [T ] is an I P -density point of [T ] w.r.t. Bor(I P ). Corollary Suppose P is a stem-linked, nondegenerate, homogeneous tree forcing on <ω. Then I P has the density property w.r.t. Bor(I P ). Philipp Schlicht Variants of Lebesgue s Density Theorem 15/19

16 Examples The I-density property w.r.t Bor(I) holds for the σ-ideals I defined by the following tree forcings, since they are stem-linked: Cohen forcing C, Hechler forcing H, F -Laver forcing L F for a filter F, and F -Mathias forcing R F for a filter F. However, random forcing B does not have a dense stem-linked subset. Philipp Schlicht Variants of Lebesgue s Density Theorem 16/19

17 Counterexamples The translation density property fails for the following tree forcings. Sacks forcing S, Mathias forcing R, Laver forcing L, Miller forcing M, and Silver forcing V. Philipp Schlicht Variants of Lebesgue s Density Theorem 17/19

18 From ideals to forcings Theorem (Ikegami) Suppose that P is a proper tree forcing. Then the map ι: P B/I P that sends T P to the I P -equivalence class represented by [T ] is a dense embedding, where B denotes the class of Borel subsets of ω and B/I P denotes the quotient Boolean algebra. A σ-ideal I on the Borel subsets of ω is homogeneous if there are I-invariant Borel isomorphisms between any two I-positive Borel sets. Philipp Schlicht Variants of Lebesgue s Density Theorem 18/19

19 Open questions Question Does the density property fail for some homogeneous ccc σ-ideal? Question Does the density property hold for some homogeneous non-ccc σ-ideal? Question Is it consistent that there is no definable selector for the equivalence relation equal modulo countable on the class of Borel sets? Philipp Schlicht Variants of Lebesgue s Density Theorem 19/19