The Joys of Haar Measure

Diestel, Joe; Spalsbury, Angela

doi:10.1090/gsm/150

Cited by 38 publications

(33 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The question of which sets have the Steinhaus-Weil property under µ, hinges on the choice of µ (see the earlier cautionary example of this section), and indeed on further delicate subcontinuity considerations, related to [Sol2], for which see [BinO9]. In this connection see [Oxt1] and [DieS,Ch. 10].…”

Section: Corollary 1 (Mospan Property [Mos Th 2]) For µ-Non-null mentioning

confidence: 99%

Beyond Lebesgue and Baire IV: Density topologies and a converse Steinhaus–Weil theorem

Bingham

Ostaszewski

2018

Topology and its Applications

View full text Add to dashboard Cite

The theme here is category-measure duality, in the context of a topological group. One can often handle the (Baire) category case and the (Lebesgue, or Haar) measure cases together, by working bi-topologically: switching between the original topology and a suitable refinement (a density topology). This prompts a systematic study of such density topologies, and the corresponding σ-ideals of negligibles. Such ideas go back to Weil's classic book, and to Hashimoto's ideal topologies. We make use of group norms, which cast light on the interplay between the group and measure structures. The Steinhaus-Weil interior-points theorem ('on AA −1 ') plays a crucial role here; so too does its converse, the Simmons-Mospan theorem.

show abstract

Section: Corollary 1 (Mospan Property [Mos Th 2]) For µ-Non-null mentioning

confidence: 99%

Beyond Lebesgue and Baire IV: Density topologies and a converse Steinhaus–Weil theorem

Bingham

Ostaszewski

2018

Topology and its Applications

View full text Add to dashboard Cite

show abstract

“…However, the metric d produced by Birkhoff's construction decreases exponentially faster than needed for our purposes due to a factor 3 2 ) n . For this, we shall instead rely on the construction of Kakutani from which a better estimate can be extracted (see [3] for a proof of the exact statement of Lemma 5 below).…”

Section: Definition 1 a Metric D On A Topological Group G Is Said Tomentioning

confidence: 99%

Lipschitz structure and minimal metrics on topological groups

Rosendal

2018

Arkiv För Matematik

View full text Add to dashboard Cite

Abstract.We discuss the problem of deciding when a metrisable topological group G has a canonically defined local Lipschitz geometry. This naturally leads to the concept of minimal metrics on G, that we characterise intrinsically in terms of a linear growth condition on powers of group elements.Combining this with work on the large scale geometry of topological groups, we also identify the class of metrisable groups admitting a canonical global Lipschitz geometry.In turn, minimal metrics connect with Hilbert's fifth problem for completely metrisable groups and we show, assuming that the set of squares is sufficiently rich, that every element of some identity neighbourhood belongs to a 1-parameter subgroup.

show abstract

“…Nothing new is obtained in our setting of probability measures, but if one drops local finiteness, Haar-like measures of 'pathological' character can occur ( §9.8 below). We quote Diestel and Spalsbury [DieS,Ch. 10], who give a textbook account of the early work of Oxtoby in this area [Oxt1].…”

Section: Complementsmentioning

confidence: 99%

“…The Th. 2], [DieS,Th.10.1]). This asserts that in a non-locally-compact Polish group carrying a (non-trivial, left) invariant Borel measure every nhd of the identity contains uncountably many disjoint (left) translates of a compact set of positive measure.…”

Section: Complementsmentioning

confidence: 99%