Chapter 44: Topological groups

441 Invariant measures on locally compact spaces
Measures invariant under group actions; Haar measures; measures invariant under isometries.

442 Uniqueness of Haar measure
Two (left) Haar measures are multiples of each other; left and right Haar measures; Haar measurable and Haar negligible sets; the modular function of a group; formulae for ∫f (x^-1)dx, ∫f (xy)dx.

443 Further properties of Haar measure
The Haar measure algebra of a group carrying Haar measures; actions of the group on the Haar measure algebra; locally compact groups; actions of the group on L⁰ and L^p; the bilateral uniformity; Borel sets are adequate; completing the group; expressing an arbitrary Haar measure in terms of a Haar measure on a locally compact group; completion regularity of Haar measure; invariant measures on the set of left cosets of a closed subgroup of a locally compact group; modular functions of subgroups and quotient groups; transitive actions of compact groups on compact spaces.

444 Convolutions
Convolutions of quasi-Radon measures; the Banach algebra of signed τ-additive measures; continuous actions and corresponding actions on L⁰(ν) for an arbitrary quasi-Radon measure ν; convolutions of measures and functions; indefinite-integral measures over a Haar measure μ; convolutions of functions; L^p(μ); approximate identities; convolution in L²(μ).

445 The duality theorem
Dual groups; Fourier-Stieltjes transforms; Fourier transforms; identifying the dual group with the maximal ideal space of L¹; the topology of the dual group; positive definite functions; Bochner's theorem; the Inversion Theorem; the Plancherel Theorem; the Duality Theorem.

446 The structure of locally compact groups
Finite-dimensional representations separate the points of a compact group; groups with no small subgroups have B-sequences; chains of subgroups.

447 Translation-invariant liftings
Translation-invariant liftings and lower densities; Vitali's theorem and a density theorem for groups with B-sequences; Haar measures have translation-invariant liftings.

448 Polish group actions
Countably full local semigroups of Aut(A); σ-equidecomposability; countably non-paradoxical groups; G-invariant additive functions from A to L^∞(C); measures invariant under Polish group actions (the Nadkarni-Becker-Kechris theorem); measurable liftings of L⁰; the Borel structure of L⁰; representing a Borel measurable action on a measure algebra by a Borel measurable action on a Polish space (Mackey's theorem).

449 Amenable groups
Amenable groups; permanence properties; the greatest ambit of a topological group; locally compact amenable groups; Tarski's theorem; discrete amenable groups; isometry-invariant extensions of Lebesgue measure.

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Revised 29.6.2016