Chapter 36: Function spaces

361 S
Additive functions on Boolean rings; the space S(A); universal mapping theorems for linear operators on S; the map T_π:S(A)→S(B) induced by a ring homomorphism π:A→B; projection bands in S(A); identifying S(A) when A is a quotient of an algebra of sets.

362 S^~
Bounded additive functionals on A identified with order-bounded linear functionals on S(A); the L-space S^~ and its bands; countably additive, completely additive, absolutely continuous and continuous functionals; uniform integrability in S^~.

363 L^∞
The space L^∞(A), as an M-space and f-algebra; universal mapping theorems for linear operators on L^∞; T_π:L^∞(A)→L^∞(B); representing L^∞ when A is a quotient of an algebra of sets; integrals with respect to finitely additive functionals; projection bands in L^∞; (L^∞)^~ and its bands; Dedekind completeness of A and L^∞; representing σ-complete M-spaces; the generalized Hahn-Banach theorem; the Banach-Ulam problem.

364 L⁰
The space L⁰(A); representing L⁰ when A is a quotient of a σ-algebra of sets; algebraic operations on L⁰; action of Borel measurable functions on L⁰; identifying L⁰(A) with L⁰(μ) when A is a measure algebra; embedding S and L^∞ in L⁰; suprema and infima in L⁰; Dedekind completeness in A and L⁰; multiplication in L⁰; projection bands; T_π:L⁰(A)→L⁰(B); when π represented by a (T,Σ)-measurable function; simple products; *regular open algebras; *the space C^∞(X).

365 L¹
The space L¹(A,μ); identification with L¹(μ); ∫_au; the Radon-Nikodým theorem again; ∫w×T_πudν=∫udμ; additive functions on A and linear operators on L¹; the duality between L¹ and L^∞; T_π:L¹(A,μ)→L¹(B,ν) and P_π:L¹(B,ν)→L¹(A,μ); conditional expectations; bands in L¹; varying μ.

366 L^p
The spaces L^p(A,μ); identification with L^p(μ); L^q as the dual of L^p; the spaces M⁰ and M^1,0; T_π:M⁰(A,μ)→M⁰(B,ν) and P_π:M^1,0(B,ν)→M^1,0(A,μ); conditional expectations; the case p=2; spaces L^p_C(A,μ).

367 Convergence in measure
Order*-convergence of sequences in lattices; in Riesz spaces; in Banach lattices; in quotients of spaces of measurable functions; in C(X); Lebesgue's Dominated Convergence Theorem and Doob's Martingale Theorem; convergence in measure in L⁰(A); and pointwise convergence; defined by the Riesz space structure; positive linear operators on L⁰; convergence in measure and the canonical projection (L¹)**→L¹; the set of independent families of random variables.

368 Embedding Riesz spaces in L⁰
Extension of order-continuous Riesz homomorphisms into L⁰; representation of Archimedean Riesz spaces as subspaces of L⁰; Dedekind completion of Riesz spaces; characterizing L⁰ spaces as Riesz spaces; weakly (σ,∞)-distributive Riesz spaces.

369 Banach function spaces
Riesz spaces separated by their order-continuous duals; representing U^× when U⊆L⁰; Kakutani's representation of L-spaces as L¹ spaces; extended Fatou norms; associate norms; L^τ′≅(L^τ)^×; Fatou norms and convergence in measure; M^∞,1 and M^1,∞, ∥ ∥_∞,1 and ∥ ∥_1,∞; L^τ1+L^τ2.

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Revised 25.8.2017