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Small Sets of Reals: Measure and Category

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Small Sets of Reals: Measure and Category 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 2= 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 2= 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; 1], called the Lebesgue measure, such that µ(;) = 0. 1 If fEi gi=0 are disjoint Lebesgue measurable sets, then 1 ! 1 [ 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 1 1 \ fOngn=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 1 1 \ fOngn=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. 1 Let Q = fqi gi=0 and let Iij be the open interval centered at qi 1 1 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. 1 Let Q = fqi gi=0 and let Iij be the open interval centered at qi 1 1 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 jz − j < 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 n 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 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 n 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 n E. Francis Adams Small Sets of Reals: Measure and Category Fubini and Kuratowski-Ulam Theorems 2 y For A ⊆ R , Ax = fy :(x; y) 2 Ag and A = fx :(x; y) 2 Ag. Let A ⊆ R2 be Lebesgue measurable. Then A is null iff y fx : Ax 2= N g 2 N iff fy : A 2= N g 2 N . Let A ⊆ R2 have BP. Then A is meager iff y fx : Ax 2= Mg 2 M iff fy : A 2= Mg 2 M. Francis Adams Small Sets of Reals: Measure and Category Erd¨os-Sierpi´nski Theorem @0 Assume the Continuum Hypothesis i.e. @1 = 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'. @0 Assume the Continuum Hypothesis i.e. @1 = 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 @0 has size @1 = 2 . But if there is a non-null set of size @1 while every nonmeager set has size > @1, 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).
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