Citation
Day, Peter William (1970) Rearrangements of Measurable Functions. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/6V2ZF375. https://resolver.caltech.edu/CaltechTHESIS:10292010132102259
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 nonatomic, then there is a measure preserving map σ: X → [0, μ(X)] such that δ_{f}(σ) = f μa.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 δ_{f} is obtained as the right inverse of 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 nonatomic and σfinite 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 μa.e.
If (X, Λ, μ) and (X_{1}, Λ_{1}, μ_{1}) are finite measure spaces such that a = μ(X) = μ_{1}(X_{1}), if f, g ϵ M(X, μ) ∪ M(X_{1}, μ_{1}), and if ∫_{o}^{a} δ_{f+} and ∫_{o}^{a} δ_{g+} are finite, then g < < f means ∫_{o}^{t} δ_{g} ≤ ∫_{o}^{t} δ_{f} for all 0 ≤ t ≤ a, and g < f means g < < f and ∫_{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 ϵ L^{1}(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 ξclosed 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_{1}, μ_{1}) is given such that Ω_{f} ⊂ L^{ρ} iff f ϵ L^{ρ}1.
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 nonatomic, 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 ∪ B) = Φ(A) ∪ Φ(B) [μ] whenever A ∩ B = Ø [μ_{1}]; and (iii) Φ(A ∩ B) = Φ(A) ∩ Φ(B)[μ] for all A, B ϵ Λ_{1}, where A = B [μ] means C_{A} = C_{B} μa.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^{1} (X_{1}, μ_{1}) → L^{1}(X, μ) is linear then Tf ~ f for all f ϵ L^{1}(X_{1}, μ_{1}) iff T = T_{Φ} for some homomorphism Φ.
Let X_{o} be the nonatomic 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 Δ(f) is shown to be {g ϵ L^{1}: there is an h ~ f such that gX_{o} < hX_{o} and gA = hA} 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, nonatomic, 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): 
 
Thesis Committee: 
 
Defense Date:  21 April 1970  
Funders: 
 
Record Number:  CaltechTHESIS:10292010132102259  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:10292010132102259  
DOI:  10.7907/6V2ZF375  
ORCID: 
 
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  
Last Modified:  07 May 2024 21:56 
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