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

245 Convergence in measure
The topology of (local) convergence in measure on L⁰; pointwise convergence; localizability and Dedekind completeness; embedding L^p 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