Real Analysis, course outline Denis Labutin 1 Measure theory I 1. Sigma algebras. Let A be a collection of subsets of some fixed set Ω . It is called a σ -algebra with the unit element Ω if (a) ;; Ω 2 A ; (b) E 2 A =) cE 2 A ; [ (c) Ej 2 A , j = 1; 2;::: =) Ej 2 A . j Prove that a σ -algebra A is closed under countable number of the set theoretic operations ( \ , [ , n , (·)c ). 2. For a sequence of sets Ej define lim sup Ej = fpoints which belong to infinitely many Ejg; j!1 lim inf Ej = fpoints which belong to all but finitely many Ejg: j!1 Thus lim sup Ej ⊃ lim inf Ej . If lim sup Ej = lim inf Ej = E , then we write lim Ej = E: j!1 (a) Give an example of a sequence fEjg for which the inclusion is strict. (b) Show that 1 0 1 1 \ [ lim sup Ej = @ EkA ; j j=1 k=j 1 0 1 1 [ \ lim inf Ej = @ Ek:A j j=1 k=j (c) Show that if all Ej are elements of a σ -algebra A , then lim sup Ej; lim inf Ej 2 A: j j (d) Suppose that E1 ⊃ E2 ⊃ · · · . Prove that \ lim sup Ej = lim inf Ej = Ej: j j j 1 (e) Find (and prove) the formula for lim sup Ej , lim inf Ej in the case E1 ⊂ E2 ⊂ · · · . 3. Prove that the following collections of sets are σ -algebras. (a) Trivial σ -algebra f;; Ωg . (b) Bulean σ -algebra 2Ω = f all subsets of Ω g . (c) For any E ⊂ Ω the collection fE; cE; ;; Ωg (It is called the σ - algebra generated by the set E ). (d) Let D = fD1;D2;:::g is a countable (disjoint) partition of Ω : [ Ω = Dj;Di \ Dj = ; if i 6= j: j The family of all at most countable unions of elements of D (includ- ing ; ) is a σ -algebra. (e) The family of all E ⊂ Ω such that either E is at most countable or cE is at most countable. 4. New σ -algebras from old. Prove the following statements. (a) Give an example showing that the union of two σ -algebras with the same unit element is not necessarily a σ -algebra. (b) If (A1; Ω) and (A2; Ω) are σ -algebras then A1 \A2 is also a σ - algebra with the same unit element Ω . \ (c) Similarly, if (Aa; Ω) is a σ -algebra for any a 2 A , then Aa is a2A also a σ -algebra with the same unit element Ω . (d) If (A; Ω) is a σ -algebra, and E ⊂ Ω , then A\ E = fA \ E : A 2 Ag is a σ -algebra with the unit E \ Ω = E . (e) If (A; Ω) is a σ -algebra, and Ξ is any fixed set, then A × Ξ = fA × Ξ: A 2 Ag is a σ -algebra with the unit Ω × Ξ . One should think of Ξ as a group of dummy variables. 5. Property (b) is the basis for the following important construction. Theorem 1 Let C be any collection of subsets of Ω . Then there exists one and only one σ -algebra (A; Ω) such that: (i) C ⊂ A . (ii) For any σ -algebra (Ae; Ω) containing C we have Ae ⊃ A . It is called the σ -algebra generated by C and is denoted by σ(C) . 2 For the proof just define def \ σ(C) = Aα: all σ−algebras (Ω;Aα): C⊂Aα Hence, the more careful notation should be σΩ(C) . 6. The definition of the σ -algebra generated by a collection of sets is a "somewhat inaccessible concept". Can we write a formula for its elements in terms of the elements of the initial collection? (a) For the collection consisting of a single E ⊂ Ω find σE(E) , σΩ(E) , and σ(fE; Ecg) . (b) For a disjoint countable partition D of Ω , [ Ω = Dj;Di \ Dj = ; if i 6= j; j find σ(D) . (c) Prove that σ(σ(C)) = σ(C) for any collection C of subsets. (d) Prove that σ(C) = σ (fF : F = Ec;E 2 Cg) 08 91 < [ = = σ @ F : F = Ej;Ej 2 C A : : j ; In general, it is not possible to give a simple formula or a simple con- structive description of σ(C) . For example, if we take all countable inter- sections and unions of open and closed interval in R1 we obtain only a proper subset of B1 . An equivalent descriptive definition of σ(C) can be given using transfinite induction. 7. If (A1; Ω1) and (A2; Ω2) are σ -algebras, define A1 × A2 = fE = E1 × E2 : Ej 2 Ajg ; and prove that A1 × A2 is not necessarily a σ -algebra. Set A1 ⊗ A2 = σ (A1 × A2) : The σ -algebra A1 ⊗ A2 with the unit element Ω1 × Ω2 is called tensor product of A1 and A2 . 8. Define the important Borel σ -algebra by writing n Bn = B(R ) = fσ − algebra generated by open setsg = fσ − algebra generated by closed setsg = fσ − algebra generated by open cubesg: 3 For a general metric (or even topological) space X its Borel σ -algebra is B(X) def= fσ − algebra generated by open subsets of Xg: n 9. The product structure of R leads to a product structure of Bn . Theorem 2 B(R2) = B(R1) ⊗ B(R1): (1.1) Proof. 1. We prover the inclusion B2 ⊂ B1 ⊗ B1: Notice that by the definition of the tensor product, the inclusion (a; b) × (c; d) 2 B1 ⊗ B1 holds for any simple open rectangle. But B2 is the minimal σ -algebra containing such simple rectangles. To prove the inclusion B2 ⊃ B1 ⊗ B1 we need to show that X × Y 2 B2 for all X; Y 2 B1 . 2. For our X; Y 2 B1 we notice that X × Y = (X × Ry) \ (Rx × Y ) : Hence, to establish the desired statement (1.1) it is sufficient to show that X × Ry; Rx × Y 2 B2: Let us prove the inclusion for the first set. Denote by I the collection of all intervals in Rx . Then X × Ry 2 B1 × Ry = σ(I) × Ry: At the same time, directly from the definitions, σ(I × Ry) ⊂ B2: Indeed, the σ -algebra in the left hand side is generated by the open rectangles of the form (a; b)×(−∞; +1) . Hence we just need to establsh that σ(I) × Ry ⊂ σ(I × Ry): (1.2) 3. Inclusion (1.2) is established using the so-called principle of good sets. This is the name for a certain type of arguments frequently used in measure theory. Define the collection of good sets to be def E = fE ⊂ R: E 2 σ(I) and E × Ry 2 σ(I × Ry)g : It is easy to check that E is a σ -algebra with the unit element Rx . Indeed, by the definition E × Ry = σ(I) × Ry \ σ(I × Ry); 4 and the intersection of σ -algebras is a σ -algebra. To prove (1.2) we just need to show that E = σ(I) . To verify this equality observe from the definitions that I ⊂ E ⊂ σ(I): Therefore σ(E) = σ(I): Since E is a σ -algebra σ(E) = E: Thus (1.2) is proved. Prove that in fact we have σ(I) × Ry = σ(I × Ry) in (1.2). 10. Maps and σ -algebras. Let f : D ! Ω , and let (A; Ω) be a σ -algebra. The full preimage f −1(A) def= f −1(E): E 2 A under f is a σ -algebra. (What is its unit element? Prove the statement.) Let (A0;D) be a σ -algebra. The direct image f(A0) = ff(E): E 2 A0g is not always a σ -algebra. (Give an example.) Define the push forward f#(A0) , −1 f#(A0) = fS ⊂ Ω: f (S) 2 A0g: Is it a σ -algebra? (Prove the statement, or give a counterexample.) 1 −1 11. A function f :Ω ! R is A -measurable if f (B1) ⊂ A . That is, −1 f (E) 2 A 8E 2 B1: Equivalently, the full preimage of the σ -algebra B(R) under f is a subalgebra of A . Prove that a function f is measurable if and only if f −1((−∞; t)) 2 A 8t: The last condition is easier to check in practice. For the proof of the "if" part one can argue using the principle of good sets. The collection of the good sets in this case is −1 E = E ⊂ R: E 2 B1 and f (E) 2 A = B1 \ f#(A): Then notice that E is a σ -algebra, E ⊂ B1 , and that (−∞; t) 2 E for any t . The statement now follows from the properties of the generating operation σ . 5 12. Similarly a complex valued function F :Ω ! C is A -measurable if −1 f (E) 2 A 8E 2 B2: −1 Equivalently, F (B2) ⊂ A (the full preimage of the σ -algebra B(C) = B2 under F is a subalgebra of A ). Prove that such F is measurable if and only if f = Re(F ) and g = Im(F ) are real valued measurable functions. 13. We will need measurable functions with the values in R [ f+1g , or R [ f+1g [ {−∞} , or C [ f1g . Hence we need to define Borel σ - algebras on these sets. 14. One point compactification of R or C is defined to be the set R = R [ f1g (or C = C [ f1g ) equipped with the following modified topology. A set O ⊂ R is open if either O ⊂ R is open, or O = R n K for some compactum K ⊂ R .