Rearrangements of Measurable Functions

Decreasing rearrangement, doubly stochastic operator, measure preserving transformation, non-atomic, measure space, extreme point, rearrangement invariant normed space, Muirhead's inequality, doubly stochastic, majorization

Day, Peter William ; Luxemburg, W. A. J.

University:California Institute of Technology

Department:Physics, Mathematics and Astronomy

关键词: Decreasing rearrangement, doubly stochastic operator, measure preserving transformation, non-atomic, measure space, extreme point, rearrangement invariant normed space, Muirhead's inequality, doubly stochastic, majorization;   

Others  :  https://thesis.library.caltech.edu/6164/1/Day_pw_1970.pdf

美国|英语

来源: Caltech THESIS

【 摘 要 】

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 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 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 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 ∫_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 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 L^ρ is universally rearrangement invariant and L^∝ ⊂ L^ρ ⊂ L¹. 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 L¹(X₁, μ₁) let Ω_f(X, μ) = {g ϵ L¹(X, μ): g < f} and Δ_f(X, μ) = {g ϵ L¹(X, μ): g ~ f}. Theorems for Ω(f) and Δ(f) are generalized to Ω_f and Δ_f, and a norm ρ₁ on M(X₁, μ₁) is given such that Ω_|f| ⊂ L^ρ iff f ϵ L^ρ1.

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 C_A = C_B μ-a.e. If Φ: Λ₁ → Λ is a homomorphism, then there is a unique doubly stochastic operator T_Φ: L¹(X₁, μ₁) → L¹ (X, μ) such that T_ΦC_E = C_Φ(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 = 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 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|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.