Chapter 24: Function spaces
L0 and L0
The linear, order and multiplicative structure of L0; Dedekind completeness and localizability; action of Borel functions.
The normed lattice L1; integration as a linear functional; completeness and Dedekind completeness; the Radon-Nikodým theorem and conditional expectations; convex functions; dense subspaces.
The normed lattice L∞; norm-completeness; the duality between L1 and L∞; localizability, Dedekind completeness and the identification L∞≅(L1)*.
The normed lattices Lp, for 1<p<∞; Hölder's inequality; completeness and Dedekind completeness; (Lp)*≅Lq; conditional expectations; *uniform convexity.
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.
Uniformly integrable sets in L1 and L1; elementary properties; disjoint-sequence characterizations; ∥ ∥1 and convergence in measure on uniformly integrable sets.
Weak compactness in L1
A subset of L1 is uniformly integrable iff it is relatively weakly compact.
ro-PDF (results-only version).
Return to contents page.