Rearrangements of Measurable Functions

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) 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 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 If f ϵ L1(X, μ), let Ω(f) = {g ϵ L1(X, μ): g A linear map T: L1(X1, μ1) → L1(X, μ) is said to be doubly stochastic iff 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 = O [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