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 *L*^{0} 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 *L*^{0}(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 *L*^{2}(μ).

445
The duality theorem

Dual groups; Fourier-Stieltjes transforms; Fourier transforms;
identifying the dual group with the maximal ideal space of *L*^{1};
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 *L*^{0}; the Borel structure of *L*^{0}; 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