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Rearrangements of measurable functions

Citation

Day, Peter William (1970) Rearrangements of measurable functions. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:10292010-132102259

Abstract

Let (X, Λ,μ) be a measure space and let M(X, μ) denote the set of all extended real valued measurable functions on X. If (X_ 1, Λ_1, μ_1) is also a measure space and f ϵ M(X, μ) and g ϵ M(X_1, μ 1), then f and g are said to be equimeasurable (written f ~ g) iff μ (f^(-1) [r, s]) = μ_1 (g ^(-1) [r, s]) whenever [r, s] is a bounded interval of the real numbers or [r, s] = {+∝} or = {-∝}. Equimeasurability is investigated systematically and in detail. If (X, Λ,μ) is a finite measure space (i. e. μ (X) < ∝) then for each f ϵ M(X, μ) the decreasing rearrangement δ_f of f is defined by δ_f(t)) = inf {s: μ ( {f > s}) ≤ t} 0 ≤ t ≤ μ (X). Then δ_f is the unique decreasing right continuous function on [0, μ (X) ] such that δ_f ~ f. If (X, Λ,μ) is non-atomic, then there is a measure preserving map σ X→ [0, μ (X) ] such that δ _f (σ) = f μ-a. e. If (X, Λ,μ) is an arbitrary measure space and f ϵ M(X, μ) then f is said to have a decreasing rearrangement iff there is an interval J of the real numbers and a decreasing function δ on J such that f ~ δ. The set D(X, μ.} of functions having decreasing rearrangements is characterized, and a particular decreasing rearrangement δ_f is defined for each f ϵ D. If ess. inf f → 0 < ess. sup f, then _f is obtained as the right inverse o£ a distribution function of f. If ess.inf f < 0 < ess.sup f then formulas relating (δ_f)+ to δ_f+, δ_f) to δ_f- and δ_-f to δ_f are given. If (X, Λ,μ) is non -atomic and σ-finite and δ is a decreasing rearrangement of f on J, then there is a measure preserving map σ: X → J such that 6 (σ) = f μ-a.e. If (X, Λ,μ) and (X_1, Λ_1,μm) are finite measure spaces such that a = μ(X) = μ_1(X_l), if f, g ϵ M(X, μ.) U M(X_1, μ_1)' and if ∫_o^a δ_f+ and ∫_o^a δ_g+ are finite, then g << f means ∫_o^a δ_g ≤ ∫_o^a δ_f for all 0 ≤ t ≤ a , and g < f means g << f ∫_o^a δ_f = ∫_o^a δ_g. The preorder relations < and << are investigated in detail. If f ϵ L^1(X, μ), let Ω (f) = {g ϵL_l(X, μ): g<f} and Δ(f) = {g ϵ L_1 (X, μ): g ~f }. Suppose ρ is a saturated Fatou norm on M(X, μ) such that L^ρ is universally rearrangement invariant and L^∝ ⊂ L^ρ ⊂ L^1. If f ϵL^ρ then Ω(f) ∩ L_ρ and Ω(f) is convex and σ(L^ρ , L^ρ')-compact. If ξ is a locally convex topology on L^ρ in which the dual of L^ρ is L_ρ', then Ω(f) is the ξ-closed convex hull of Δ(f) for all f ϵ L^ρ iff (X, Λ, μ) is adequate. More generally, if f ϵ L^1 (X_1, μ_1) let Ω_f(X, μ) = {g ϵ L^1 (X, μ): g < f} and Δ_f(X,μ) = {g ϵ L^1 (X, μ): g ~ f}. Theorems for Ω(f) and Δ(f) are generalized to Ω_f and Δ_f, and a norm ρ_1 on M(X_l, μ_1) is given such that Ω _|f| ⊂ L^ρ iff f ϵ L^ρl. A linear map T: L_1(X_1,μ_1) →L^1(X,μ) is said to be doubly stochastic iff Tf<f for all f ϵ L^1(X_1,μ_1). It is shown that g < f. iff there is a doubly stochastic T such that g = Tf. If f ϵ L^1 then the members of Δ (f) are always extreme in Ω(f). If (X, Λ, μ) is non-atomic, then Δ(f) is the set of extreme points and the set of exposed points of Ω(f). A mapping Φ: Λ_1 →Λ is called a homomorphism if (i) μ Φ (A)) = μ_1 (A) for all A ϵ Λ_1; (ii) Φ (A U B) = Φ (A) U Φ (B) [μ] whenever A ∩ B . = Ø [μ_1]; and (iii) Φ (A ∩ B) [μ] for all A, B ϵ Λ_1, where A = B [μ] means C_A = C_B μ-a.e. If Φ: Λ_1 →Λ is a homomorphism, then there is a unique doubly stochastic operator T Φ: L^1 (X_1, μ_1)→L^1 (X, μ.) such that T Φ C_E = C_ Φ _ (E) for all E. If T:L^I (X_1,μ_1)→L^1(X,μ) is linear then Tf ~ f for all f ϵ L^l'(X_ 1,μ_1) iff T = T Φ for some homomorphism Φ. Let X_o be the non-atomic part of X and let A be the union of the atoms of X. If f ϵ L^1 (X,μ) then the σ(L^1,L^∝) - closure of Λ(f) is shown to be {g ϵ^1: there is an h ~ f such that g|X_o < h|X_o and g|A = h|A} whenever either (i) X consists only of atoms; (ii) X has only finitely many atoms; or (iii) X is separable.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:decreasing rearrangement, doubly stochastic operator, measure preserving transformation, non-atomic, measure space, extreme point, rearrangement invariant normed space, Muirhead's inequality, doubly stochastic, majorization
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Luxemburg, W. A. J.
Thesis Committee:
  • Unknown, Unknown
Defense Date:21 April 1970
Record Number:CaltechTHESIS:10292010-132102259
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:10292010-132102259
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:6164
Collection:CaltechTHESIS
Deposited By: John Wade
Deposited On:29 Oct 2010 21:44
Last Modified:26 Dec 2012 04:32

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