Chapter 39: Measurable algebras

391
Kelley's theorem

Measurable algebras; strictly positive additive functionals and weak
(σ,∞)-distributivity; additive functionals subordinate
to or dominating a given functional; intersection numbers;
existence of strictly positive additive functionals.

392
Submeasures

Submeasures; exhaustive, uniformly exhaustive and Maharam
submeasures; the Kalton-Roberts theorem (a strictly positive
uniformly exhaustive submeasure provides a strictly positive additive
functional); strictly positive submeasures, associated metrics and
metric completions of algebras; products of submeasures.

393
Maharam algebras

Maharam submeasures; Maharam algebras; topologies on Boolean
algebras; order-sequential topologies; characterizations of Maharam
algebras.

394
Talagrand's example

PV norms; exhaustive submeasures which are not uniformly exhaustive;
non-measurable Maharam algebras; control measures.

395
Kawada's theorem

Full local semigroups; τ-equidecomposability; fully
non-paradoxical subgroups of Aut(**A**); ⎣b:a⎦ and
⌈b:a⌉; invariant additive functions from **A** to
*L*^{∞}(**C**), where **C** is the fixed-point
subalgebra of a group; invariant additive functionals and measures;
ergodic fully non-paradoxical groups.

396
The Hajian-Ito theorem

Invariant measures on measurable algebras; weakly wandering elements.

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Revised 2.9.2018