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
*L*^{0}

The space *L*^{0}(**A**); representing *L*^{0} when **A** is a
quotient of a σ-algebra of sets; algebraic operations on
*L*^{0}; action of Borel measurable functions on *L*^{0}; identifying
*L*^{0}(**A**) with *L*^{0}(μ) when **A** is a measure algebra;
embedding *S* and *L*^{∞} in *L*^{0}; suprema and infima in
*L*^{0}; Dedekind completeness in **A** and *L*^{0}; multiplication
in *L*^{0}; projection bands;
*T*_{π}:*L*^{0}(**A**)→*L*^{0}(**B**); when π represented by
a (T,Σ)-measurable function; simple products; *regular
open algebras; *the space *C*^{∞}(*X*).

365
*L*^{1}

The space *L*^{1}(**A**,μ); identification with *L*^{1}(μ);
∫_{a}*u*; the Radon-Nikodým theorem again;
∫*w*×*T*_{π}*ud*ν=∫*ud*μ; additive
functions on **A** and linear operators on *L*^{1}; the duality
between *L*^{1} and *L*^{∞};
*T*_{π}:*L*^{1}(**A**,μ)→*L*^{1}(**B**,ν) and
*P*_{π}:*L*^{1}(**B**,ν)→*L*^{1}(**A**,μ);
conditional expectations; bands in *L*^{1}; varying μ.

366
*L*^{p}

The spaces *L*^{p}(**A**,μ); identification with *L*^{p}(μ);
*L** ^{q}* as the dual of

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 *L*^{0}(**A**); and
pointwise convergence; defined by the Riesz space structure;
positive linear operators on *L*^{0}; convergence in measure and the
canonical projection (*L*^{1})**→*L*^{1}; the set of independent
families of random variables.

368
Embedding Riesz spaces in *L*^{0}

Extension of order-continuous Riesz homomorphisms into *L*^{0};
representation of Archimedean Riesz spaces as subspaces of *L*^{0};
Dedekind completion of Riesz spaces; characterizing *L*^{0} 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*Í*L*^{0}; Kakutani's representation of
*L*-spaces as *L*^{1} 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