Assignment #5

PMATH 451/651, FALL 2015 Assignment #5

Due: Fri. Nov. 27.

1. Define, for a Borel set E ⊂ Rd and x in Rd , the upper and lower densities DE (x) = lim sup r→0+

λ(E ∩ Br (x) λ(E ∩ Br(x) , DE (x) = lim inf r→0+ λ(Br (x)) λ(Br (x))

where λ = λd is the Lebesgue measure. (a) Show that for λ-a.e. x in E we have DE (x) = DE (x) = 1, while for λ-a.e. x in Rd \ E we have DE (x) = DE (x) = 0. (b) Construct, for any pair 0 < α ≤ β < 1, a Borel set E ⊂ Rd and a point x for which DE (x) = α while DE (x) = β. [You may choose d = 1 or d = 2, if you wish.] (Bonus) Obtain (b) as above, but with choices 0 = α < β = 1, 0 = α < β < 1 and 0 < α < β = 1. 2. All aspects of this question work in Rd . For simplicity of notation, let us work with d = 2. (a) Show that the family of intervals E = {(a1 , b1 ] × (a2 , b2 ] : a1 ≤ b1 , a2 ≤ b2 in R} generates B(R2 ). (b) Show that if µ : B(R2 ) → [0, ∞] is a locally finite measure, then µ is outer regular. [Hint: Let µ0 = µ|A : A = hEi → [0, ∞]. In the notation of A1 Q4 (a), show that Aσδ = Gδ .] 3. (a) Let F ∈ BVr , TF : R → [0, ∞) the total variation function of F µF : B(R) → C the associated measure, and |µF | its total variation measure (as defined in A4 Q5). Show that |µF | = µTF . (b) Deduce that the two notions of Jordan decomposition coincide, i.e. ± Reµ± F = µTReF ±ReF and ImµF = µTImF ±ImF . Remark: This, along with the characterization of locally finite positive measures on B(R), shows that for F ∈ BVr there is a unique measure µF satisfying µF ((a, b]) = F (b) − F (a). In particular, every complex measure on B(R) is of the form µF for some F in BVr . 1

 4. (a) Let (X, M, ν) be a complex measure space for which {x} : x ∈ X ⊂ M. Show that Pthe set of discrete points Dν = {x ∈ X : ν({x}) = 6 0} satisfies x∈Dν |ν({x})| < ∞. Hence show that νd = P x∈Dν ν({x})δx defines a complex measure, and that νd ⊥ νc , where νc = ν − νd . We call νd the discrete part and νc the continuous part of ν. (b) Let F ∈ BVr . Show that the P following decomposition holds: F = Fd + Fcs + Fac ; where Fd = a∈DF F (a)Ha is a sum of Heaviside functions (Ha = χ[a,∞) ) over the set DF of discontinuities of F ; Fsc is continuous, with Fsc0 = 0 λ-a.e. (the singular-continuous part); and Fac is absolutely continuous. Moreover, each Fd , Fsc , Fac ∈ BVr with TF = TFd + TFsc + TFac . 5. (Containment relations of Lp -spaces.) Suppose 1 ≤ p < q ≤ ∞. (a) Show that Lp (µ) 6⊂ Lq (µ) ⇔ (M, µ) admits sets of arbitrary small positive measure. (b) Show that Lq (µ) 6⊂ Lp (µ) ⇔ (M, µ) admits sets of arbitrary large finite measure. 6. (Geometry of Lp spaces.) Let (L, k · k) be a normed space. A point f in L with kf k = 1 is called an extreme point of the unit ball if f = (1 − t)f1 + tf2 , 0 < t < 1 and kf1 k, kf2 k ≤ 1

⇒

f1 = f2

i.e. f cannot be represented as a proper convex combination of other elements of the unit ball. Let (X, M, µ) be a measure space. (a) Let 1 < p < ∞. Show that every norm 1 element of the unit ball of Lp (µ) is an extreme point. (b) An atom for µ is a set A in M such that µ(A) > 0 and any subset E ⊆ A, E ∈ M satisfies µ(E) = µ(A) or µ(E) = 0. Show that the unit ball of L1 (µ) admits an extreme point if and only if µ admits atom of finite measure.

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