Contents of Measure Theory, by D.H.Fremlin

Chapter 37: Linear operators between function spaces

371 The Chacon-Krengel theorem
L~(U,V)=L×(U;V)=B(U;V) for L-spaces U and V; the class T(0)μν of ∥ ∥1-decreasing, ∥ ∥-decreasing linear operators from M1,0(A,μ) to M1,0(B,μ).

372 The ergodic theorem
The Maximal Ergodic Theorem and the Ergodic Theorem for operators in T(0)μμ; for inverse-measure-preserving functions φ:XX; limit operators as conditional expectations; applications to continued fractions; mixing and ergodic transformations.

373 Decreasing rearrangements
The classes T, T×; the space M0,∞; decreasing rearrangements u*; ∥u*∥p=∥up; ∫|Tu×v|≤∫uv* if TT; the very weak operator topology and compactness of T; v is expressible as Tu, where TT, iff ∫0tv*≤∫0tu* for every t; finding T such that ∫Tu×v=∫uv*; the adjoint operator from T(0)μν to T(0)νμ.

374 Rearrangement-invariant spaces
T-invariant subspaces of M1,∞, and T-invariant extended Fatou norms; relating T-invariant norms on different spaces; rearrangement-invariant sets and norms; when rearrangement-invariance implies T-invariance.

375 Kwapien's theorem
Linear operators on L0 spaces; if B is measurable, a positive linear operator from L0(A) to L0(B) can be assembled from Riesz homomorphisms.

376 Integral operators
Kernel operators; free products of measure algebras and tensor products of L0 spaces; tensor products of L1 spaces; abstract integral operators (i) as a band in L×(U,V) (ii) represented by kernels belonging to L0(A)⊗L0(B) (iii) as operators converting weakly convergent sequences into order*-convergent sequences; operators into M-spaces or out of L-spaces.

377 Function spaces of reduced products
Measure-bounded Boolean homomorphisms on products of probability algebras; associated maps on subspaces of ∏i∈IL0(Ai) and ∏i∈ILp(Ai); reduced powers; universal mapping theorems for function spaces on projective and inductive limits of probability algebras.

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Revised 10.4.10