Contents of Measure Theory, by D.H.Fremlin

Chapter 24: Function spaces

241 L0 and L0
The linear, order and multiplicative structure of L0; Dedekind completeness and localizability; action of Borel functions.

242 L1
The normed lattice L1; integration as a linear functional; completeness and Dedekind completeness; the Radon-Nikodým theorem and conditional expectations; convex functions; dense subspaces.

243 L
The normed lattice L; norm-completeness; the duality between L1 and L; localizability, Dedekind completeness and the identification L≅(L1)*.

244 Lp
The normed lattices Lp, for 1<p<∞; Hölder's inequality; completeness and Dedekind completeness; (Lp)*≅Lq; conditional expectations; *uniform convexity.

245 Convergence in measure
The topology of (local) convergence in measure on L0; pointwise convergence; localizability and Dedekind completeness; embedding Lp in L0; ∥ ∥1-convergence and convergence in measure; σ-finite spaces, metrizability and sequential convergence.

246 Uniform integrability
Uniformly integrable sets in L1 and L1; elementary properties; disjoint-sequence characterizations; ∥ ∥1 and convergence in measure on uniformly integrable sets.

247 Weak compactness in L1
A subset of L1 is uniformly integrable iff it is relatively weakly compact.

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Revised 30.8.13