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# Rearrangements of Measurable Functions

## Citation

Day, Peter William (1970) Rearrangements of Measurable Functions. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/6V2Z-F375. https://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 (X1, Λ1, μ1) is also a measure space and f ϵ M(X, μ) and g ϵ M(X1, μ1), then f and g are said to be equimeasurable (written f ~ g) iff μ (f-1[r, s]) = μ1(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 δf of f is defined by

δf(t) = inf {s: μ ({f > s}) ≤ t} 0 ≤ t ≤ μ(X).

Then δf is the unique decreasing right continuous function on [0, μ(X)] such that δf ~ f. If (X, Λ, μ) is non-atomic, then there is a measure preserving map σ: X → [0, μ(X)] such that δf(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 δf is defined for each f ϵ D. If ess. inf f ≤ 0 < ess. sup f, then δf is obtained as the right inverse of a distribution function of f. If ess. inf f < 0 < ess. sup f then formulas relating (δf)+ to δf+, (δf)- to δf- and δ-f to δf 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 δ(σ) = f μ-a.e.

If (X, Λ, μ) and (X1, Λ1, μ1) are finite measure spaces such that a = μ(X) = μ1(X1), if f, g ϵ M(X, μ) ∪ M(X1, μ1), and if ∫oa δf+ and ∫oa δg+ are finite, then g < < f means ∫ot δg ≤ ∫ot δf for all 0 ≤ t ≤ a, and g < f means g < < f and ∫oa δf = ∫oa δg. The preorder relations < and < < are investigated in detail.

If f ϵ L1(X, μ), let Ω(f) = {g ϵ L1(X, μ): g < f} and Δ(f) = {g ϵ L1(X, μ): g ~ f}. Suppose ρ is a saturated Fatou norm on M(X, μ) such that Lρ is universally rearrangement invariant and L ⊂ Lρ ⊂ L1. If f ϵLρ then Ω(f) ⊂ Lρ and Ω(f) is convex and σ(Lρ, Lρ')-compact. If ξ is a locally convex topology on Lρ in which the dual of Lρ is Lρ', then Ω(f) is the ξ-closed convex hull of Δ(f) for all f ϵ Lρ iff (X, Λ, μ) is adequate. More generally, if f ϵ L1(X1, μ1) let Ωf(X, μ) = {g ϵ L1(X, μ): g < f} and Δf(X, μ) = {g ϵ L1(X, μ): g ~ f}. Theorems for Ω(f) and Δ(f) are generalized to Ωf and Δf, and a norm ρ1 on M(X1, μ1) is given such that Ω|f| ⊂ Lρ iff f ϵ Lρ1.

A linear map T: L1(X1, μ1) → L1(X, μ) is said to be doubly stochastic iff Tf < f for all f ϵ L1(X1, μ1). It is shown that g < f iff there is a doubly stochastic T such that g = Tf.

If f ϵ L1 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: Λ1 → Λ is called a homomorphism if (i) μ(Φ(A)) = μ1(A) for all A ϵ Λ1; (ii) Φ(A ∪ B) = Φ(A) ∪ Φ(B) [μ] whenever A ∩ B = Ø [μ1]; and (iii) Φ(A ∩ B) = Φ(A) ∩ Φ(B)[μ] for all A, B ϵ Λ1, where A = B [μ] means CA = CB μ-a.e. If Φ: Λ1 → Λ is a homomorphism, then there is a unique doubly stochastic operator TΦ: L1(X1, μ1) → L1 (X, μ) such that TΦCE = CΦ(E) for all E. If T: L1 (X1, μ1) → L1(X, μ) is linear then Tf ~ f for all f ϵ L1(X1, μ1) iff T = TΦ for some homomorphism Φ.

Let Xo be the non-atomic part of X and let A be the union of the atoms of X. If f ϵ L1(X, μ) then the σ(L1, L)-closure of Δ(f) is shown to be {g ϵ L1: there is an h ~ f such that g|Xo < h|Xo 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:(Mathematics) ; 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)
• Luxemburg, W. A. J.
Thesis Committee:
• Unknown, Unknown
Defense Date:21 April 1970
Funders:
Funding AgencyGrant Number
NSFUNSPECIFIED
Record Number:CaltechTHESIS:10292010-132102259
Persistent URL:https://resolver.caltech.edu/CaltechTHESIS:10292010-132102259
DOI:10.7907/6V2Z-F375
ORCID:
AuthorORCID
Day, Peter William0000-0002-8076-4048
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:6164
Collection:CaltechTHESIS
Deposited By:INVALID USER
Deposited On:29 Oct 2010 21:44