Small Sets of Reals: Measure and Category

Francis Adams

August 31, 2016

Francis Adams Small Sets of Reals: Measure and Category Question

What sets of reals can be considered small, and how do diﬀerent notions of smallness compare?

Francis Adams Small Sets of Reals: Measure and Category Countable Sets

Countable sets must be considered small. The countable union of countable sets is countable. A subset of a countable set is countable. R isn’t countable.

Francis Adams Small Sets of Reals: Measure and Category What are some other interesting σ-ideals containing the countable sets?

σ-ideals

Deﬁnition A collection I of subsets of R is a σ-ideal if R ∈/ I , I is closed under taking subsets, and I is closed under taking countable unions.

So the collection of countable sets is a σ-ideal on R.

Francis Adams Small Sets of Reals: Measure and Category σ-ideals

Deﬁnition A collection I of subsets of R is a σ-ideal if R ∈/ I , I is closed under taking subsets, and I is closed under taking countable unions.

So the collection of countable sets is a σ-ideal on R. What are some other interesting σ-ideals containing the countable sets?

Francis Adams Small Sets of Reals: Measure and Category The σ-algebra generated by the open sets is called the Borel σ-algebra. open and closed sets

Gδ (countable intersection of open sets) and Fσ (countable union of closed sets)

Gδσ (countable union of Gδ sets) and Fσδ (countable intersection of Fσ sets) Keep going to get a hierarchy of uncountable length.

Null Ideal

A σ-algebra on the reals is a collection of subsets closed under countable unions and complements (so also under countable intersections).

Francis Adams Small Sets of Reals: Measure and Category Null Ideal

A σ-algebra on the reals is a collection of subsets closed under countable unions and complements (so also under countable intersections).

The σ-algebra generated by the open sets is called the Borel σ-algebra. open and closed sets

Gδ (countable intersection of open sets) and Fσ (countable union of closed sets)

Gδσ (countable union of Gδ sets) and Fσδ (countable intersection of Fσ sets) Keep going to get a hierarchy of uncountable length.

Francis Adams Small Sets of Reals: Measure and Category Null Ideal

There is a σ-algebra Σ containing the Borel sets, called the Lebesgue measurable sets, and a function µ :Σ → [0, ∞], called the Lebesgue measure, such that µ(∅) = 0. ∞ If {Ei }i=0 are disjoint Lebesgue measurable sets, then ∞ ! ∞ [ X µ Ei = µ(Ei ). i=0 i=0 µ((a, b)) = b − a. If µ(A) = 0 and B ⊆ A, then µ(B) = 0.

Francis Adams Small Sets of Reals: Measure and Category Null Ideal

The collection N of sets with Lebesgue measure zero is a σ-ideal on R containing the countable sets.

Francis Adams Small Sets of Reals: Measure and Category Deﬁnition A set of reals is meager if it can be covered by countably many closed nowhere dense sets. A set is nonmeager if it is not meager, and comeager if its complement is meager (so it contains a dense Gδ).

Meager Ideal

Recall the Baire Category Theorem: R is not the union of countably many closed, nowhere dense sets. More precisely, if ∞ ∞ \ {On}n=0 are open dense sets in R then On is dense. n=0

Francis Adams Small Sets of Reals: Measure and Category Meager Ideal

Recall the Baire Category Theorem: R is not the union of countably many closed, nowhere dense sets. More precisely, if ∞ ∞ \ {On}n=0 are open dense sets in R then On is dense. n=0 Deﬁnition A set of reals is meager if it can be covered by countably many closed nowhere dense sets. A set is nonmeager if it is not meager, and comeager if its complement is meager (so it contains a dense Gδ).

Francis Adams Small Sets of Reals: Measure and Category Meager Ideal

The collection M of meager sets is a σ-ideal on R containing the countable sets. M is generated by the closed nowhere dense sets.

Francis Adams Small Sets of Reals: Measure and Category But they aren’t the same; R can be decomposed into a null set and a meager set. ∞ Let Q = {qi }i=0 and let Iij be the open interval centered at qi ∞ ∞ 1 [ \ with radius 2i+j+1 . Let Gj = Iij and let N = Gj . Then i=0 j=0 1 µ(Gj ) ≤ 2j so µ(N) = 0. But each Gj is open dense, so N is a dense Gδ.

M and N

These ideals contain uncountable sets, e.g. the Cantor set.

Francis Adams Small Sets of Reals: Measure and Category M and N

These ideals contain uncountable sets, e.g. the Cantor set.

But they aren’t the same; R can be decomposed into a null set and a meager set. ∞ Let Q = {qi }i=0 and let Iij be the open interval centered at qi ∞ ∞ 1 [ \ with radius 2i+j+1 . Let Gj = Iij and let N = Gj . Then i=0 j=0 1 µ(Gj ) ≤ 2j so µ(N) = 0. But each Gj is open dense, so N is a dense Gδ.

Francis Adams Small Sets of Reals: Measure and Category Liouville Numbers

Deﬁnition A real z is a Liouville number if z is irrational and for each n > 0 there are integers p, q such that q > 1 and p 1 |z − | < q qn

Every Liouville number is transcendental. The set of Liouville numbers has measure zero, but is a dense Gδ.

Francis Adams Small Sets of Reals: Measure and Category Similarities between M and N

Say a set has the Baire Property if it is in the σ-algebra generated by the open sets and the meager sets.

A set A has BP iﬀ A = P4M where P is open and M is meager.

A = G ∪ M wehre G is Gδ and M is meager.

A = F \ M where F is Fσ and M is meager.

Francis Adams Small Sets of Reals: Measure and Category Similarities between M and N

A set A is measurable iﬀ

A = F ∪ N wehre F is Fσ and N is null.

A = G \ N where G is Gδ and N is null.

So a set is Lebesgue measurable iﬀ it is in the σ-algebra generated by the open sets and the null sets.

Francis Adams Small Sets of Reals: Measure and Category Similarities between M and N

If every subset of E ⊆ R is measurable, then E is null. If every subset of E ⊆ R has BP, then E is meager.

Francis Adams Small Sets of Reals: Measure and Category If f is Baire measurable, then f is continuous on a comeager set. If f is Lebesgue measurable, then for every > 0 there is E ⊆ R such that µ(E) < and f is continuous on R \ E.

Measureable Functions are Almost Continuous

A function f : R → R is Baire measurable if if f −1(U) has BP for any open U. A function f : R → R is Lebesgue measurable if f −1(U) is Lebesgue measurable for any open U.

Francis Adams Small Sets of Reals: Measure and Category Measureable Functions are Almost Continuous

A function f : R → R is Baire measurable if if f −1(U) has BP for any open U. A function f : R → R is Lebesgue measurable if f −1(U) is Lebesgue measurable for any open U. If f is Baire measurable, then f is continuous on a comeager set. If f is Lebesgue measurable, then for every > 0 there is E ⊆ R such that µ(E) < and f is continuous on R \ E.

Francis Adams Small Sets of Reals: Measure and Category Fubini and Kuratowski-Ulam Theorems

2 y For A ⊆ R , Ax = {y :(x, y) ∈ A} and A = {x :(x, y) ∈ A}.

Let A ⊆ R2 be Lebesgue measurable. Then A is null iff y {x : Ax ∈/ N } ∈ N iff {y : A ∈/ N } ∈ N . Let A ⊆ R2 have BP. Then A is meager iff y {x : Ax ∈/ M} ∈ M iff {y : A ∈/ M} ∈ M.

Francis Adams Small Sets of Reals: Measure and Category Erd¨os-Sierpi´nski Theorem

ℵ0 Assume the Continuum Hypothesis i.e. ℵ1 = 2 . Then there exists a bijection f : R → R such that f = f −1 and E is meager iﬀ f (E) is null. E is null iﬀ f (E) is meager.

Francis Adams Small Sets of Reals: Measure and Category Erd¨os-Sierpi´nski Theorem

This suggests the following ’Duality Principle’.

ℵ0 Assume the Continuum Hypothesis i.e. ℵ1 = 2 . Let P be a proposition involving just set theoretical concepts as well as the notions of ’null set’ and ’meager set’. Let P∗ be the proposition obtained from P by interchanging ’null set’ and ’meager set’ wherever they appear. Then P and P∗ are equivalent.

Francis Adams Small Sets of Reals: Measure and Category Erd¨os-Sierpi´nski Theorem

CH implies that every non-null set and every nonmeager set ℵ0 has size ℵ1 = 2 . But if there is a non-null set of size ℵ1 while every nonmeager set has size > ℵ1, Erd¨os-Sierpi´nskifails.

Francis Adams Small Sets of Reals: Measure and Category Independence

It is well known that the parallel postulate is independent of the other 4 axioms of geometry (Right angles are congruent, etc.) This means there are geometries where all 5 axioms are true (Euclidean) and there are geometries where the parallel postulate is false while the other 4 are true (non-Euclidean).

Francis Adams Small Sets of Reals: Measure and Category Independence

A sentence φ is independent of a set of sentences Γ if there is a model where each sentence of Γ is true and φ is also true (φ is consistent with Γ) there is a model where each sentence of Γ is true and φ is false (¬φ is consistent with Γ)

Francis Adams Small Sets of Reals: Measure and Category Example of Independence

Let Γ be the axioms for linear orders: ∀x ¬(x < x), ∀x, y x < y ∨ y < x ∨ x = y, ∀x, y, z (x < y ∧ y < z) → x < z. A linear order L = (L, <) is a model of Γ. The sentence ∀x, y ∃z (x < z < y) ∨ (y < z < x) is independent of Γ. Q is a dense linear order, Z isn’t.

Francis Adams Small Sets of Reals: Measure and Category Another Example of Independence

Let Γ be the axioms of group theory: ∀x x · x −1 = e, ∀x x · e = e · x = x, ∀x, y, z x · (y · z) = (x · y) · z. A group G = (G, ·, −1, e) is a model of Γ. The sentence ∀x, y x · y = y · x is independent of Γ. That is, there are abelian groups and non-abelian groups.

Francis Adams Small Sets of Reals: Measure and Category Independence in Set Theory

A model of set theory is (V , ∈) satisfying ZFC (Zermelo-Fraenkel with Choice) (Extensionality) If x and y have the same elements, then x = y. (Foundation) If x 6= ∅, then there is a y ∈ x such that x ∩ y = ∅. (Comprehension) If x is a set and φ(u) is a formula, the set y = {z ∈ x : φ(z)} exists. (Pairing) If x, y are sets, then {x, y} exists. (Union) If x is a set, the set S x which is the union of all elements of x exists. (Replacement) If φ(u, v) is a formula such that for every x there is a unique y with φ(x, y), then {y : ∃x φ(x, y)} exists. (Inﬁnity) There exists an inﬁnite set. (Power Set) For any set x, its power set P(x) exists. (Choice) For any family of nonempty sets, there is a choice function.

Francis Adams Small Sets of Reals: Measure and Category Independence in Set Theory

CH is independent of ZFC, so there are models of ZFC where ℵ0 ℵ0 ℵ1 = 2 and there are models of ZFC where ℵ1 < 2 .

Francis Adams Small Sets of Reals: Measure and Category ℵ0 We can show that ℵ1 ≤ add(I ) ≤ cov(I ) ≤ cof (I ) ≤ 2 ℵ0 and that ℵ1 ≤ add(I ) ≤ non(I ) ≤ cof (I ) ≤ 2 .

Ideal Invariants

For a σ-ideal I on R, we can deﬁne the following cardinal numbers associated with it. add(I ) = least size of a family of members of I whose union is not in I . cov(I ) = least size of a family of members of I whose union is R. non(I ) = least size of a subset of R not in I cof (I ) = least size of a family B of members of I such that every member of I is a subset of some element of B.

Francis Adams Small Sets of Reals: Measure and Category Ideal Invariants

ℵ0 We can show that ℵ1 ≤ add(I ) ≤ cov(I ) ≤ cof (I ) ≤ 2 ℵ0 and that ℵ1 ≤ add(I ) ≤ non(I ) ≤ cof (I ) ≤ 2 .

Francis Adams Small Sets of Reals: Measure and Category Invariants for M and N

How these cardinal invariants compare for the meager and null ideals is encapsulated in the Cichon diagram.

In this diagram, cardinals increase as you go up/right.

Francis Adams Small Sets of Reals: Measure and Category Invariants for M and N

Any inequality not shown in the diagram is consistently false and every provable inequality is consistently strict. So it is consistent that non(M) = ℵ1 and cov(M) > ℵ1 (so also non(N ) > ℵ1). In this model the Erd¨os-Sierpi´nski theorem fails.

Francis Adams Small Sets of Reals: Measure and Category Thanks!

Figure from: Diego Alejandro Mejia, Template iterations with non-deﬁnable ccc forcing notion. For further reading: John Oxtoby, Measure and Category. Bartoszynski and Judah, On the Structure of the Real Line.

Francis Adams Small Sets of Reals: Measure and Category