Chapter 49: Further topics
The asymptotic density ideal Z; equidistributed sequences; when equidistributed sequences exist; Z=PN/Z; effectively regular measures; equidistributed sequences and induced embeddings of measure algebras in Z.
Combinatorial concentration of measure
The Hamming metric; concentration of measure in product spaces; concentration of measure in permutation groups.
Extremely amenable groups
Extremely amenable groups; concentrating additive functionals; measure algebras under Δ; L0; isometry groups of spheres in inner product spaces; locally compact groups.
Groups of measure-preserving automorphisms
Weak and uniform topologies on AutμA; a weakly mixing automorphism which is not mixing; full subgroups and fixed-point subalgebras; extreme amenability; automatic continuity; algebraic cofinality.
Poisson point processes
Poisson distributions; Poisson point processes; disintegrations; transforming disjointness into stochastic independence; representing Poisson point processes by Radon measures; exponential distributions and Poisson point processes on [0,∞[.
Submeasures; totally finite Radon submeasures; Souslin's operation; (K-)analytic spaces; product submeasures.
Tao's proof of Szemerédi's theorem
Τ-removable intersections; and relative independence; permutation-invariant measures on P([I]<ω); and Τ-removable intersections; the Hypergraph Removal Lemma; Szemerédi's theorem; a multiple recurrence theorem.
Cubes in product spaces
Subsets of measure algebras with non-zero infima; product sets included in given sets of positive measure.
ro-PDF (results-only version).
Return to contents page.