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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.)) decreasing rearrangement, doubly stochastic operator, measure preserving transformation, non-atomic, measure space, extreme point, rearrangement invariant normed space, Muirhead's inequality, doubly stochastic, majorization California Institute of Technology Physics, Mathematics and Astronomy Mathematics Public (worldwide access) Luxemburg, W. A. J. Unknown, Unknown 21 April 1970 CaltechTHESIS:10292010-132102259 http://resolver.caltech.edu/CaltechTHESIS:10292010-132102259 No commercial reproduction, distribution, display or performance rights in this work are provided. 6164 CaltechTHESIS John Wade 29 Oct 2010 21:44 22 Aug 2016 21:21

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