Chapter 24: Function spaces

241
*L*^{0} and *L*^{0}

The linear, order and multiplicative structure of *L*^{0}; Dedekind
completeness and localizability; action of Borel functions.

242
*L*^{1}

The normed lattice *L*^{1}; 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 *L*^{1} and *L*^{∞}; localizability, Dedekind
completeness and the identification *L*^{∞}≅(*L*^{1})*.

244
*L*^{p}

The normed lattices *L** ^{p}*, for 1<p<∞; Hölder's inequality;
completeness and Dedekind completeness; (

245
Convergence in measure

The topology of (local) convergence in measure on *L*^{0}; pointwise
convergence; localizability and Dedekind completeness; embedding
*L** ^{p}* in

246
Uniform integrability

Uniformly integrable sets in *L*^{1} and *L*^{1}; elementary
properties; disjoint-sequence characterizations; ∥ ∥_{1} and
convergence in measure on uniformly integrable sets.

247
Weak compactness in *L*^{1}

A subset of *L*^{1} is uniformly integrable iff it is relatively weakly
compact.

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Revised 30.8.13