Chapter 36: Function spaces
Additive functions on Boolean rings; the space S(A); universal mapping theorems for linear operators on S; the map Tπ: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.
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~.
The space L∞(A), as an M-space and f-algebra; universal mapping theorems for linear operators on L∞; Tπ: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.
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; Tπ:L0(A)→L0(B); when π represented by a (T,Σ)-measurable function; simple products; *regular open algebras; *the space C∞(X).
The space L1(A,μ); identification with L1(μ); ∫au; the Radon-Nikodým theorem again; ∫w×Tπudν=∫udμ; additive functions on A and linear operators on L1; the duality between L1 and L∞; Tπ:L1(A,μ)→L1(B,ν) and Pπ:L1(B,ν)→L1(A,μ); conditional expectations; bands in L1; varying μ.
The spaces Lp(A,μ); identification with Lp(μ); Lq as the dual of Lp; the spaces M0 and M1,0; Tπ:M0(A,μ)→M0(B,ν) and Pπ:M1,0(B,ν)→M1,0(A,μ); conditional expectations; the case p=2; spaces LpC(A,μ).
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.
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.
Banach function spaces
Riesz spaces separated by their order-continuous duals; representing U× when UÍL0; Kakutani's representation of L-spaces as L1 spaces; extended Fatou norms; associate norms; Lτ′≅(Lτ)×; Fatou norms and convergence in measure; M∞,1 and M1,∞, ∥ ∥∞,1 and ∥ ∥1,∞; Lτ1+Lτ2.
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