## Abstract

A Polish group G is called a group of quasi-invariance or a QI-group, if there exist a locally compact group X and a probability measure μ on X such that (1) there exists a continuous monomorphism ø from G into X with dense image, and (2) for each gεX either gεø(G) and the shift μg is equivalent to μ or g∉ø(G) and μg is orthogonal to μ. It is proved that ø(G) is a σ-compact subset of X. We show that there exists a Polish non-locally quasi-convex (and hence nonreflexive) QI-group such that its bidual is not a QI-group. It is proved also that the bidual group of a QI-group may be not a saturated subgroup of X. It is constructed a reflexive non-discrete group topology on the integers.

Original language | English |
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Pages (from-to) | 2786-2802 |

Number of pages | 17 |

Journal | Topology and its Applications |

Volume | 157 |

Issue number | 18 |

DOIs | |

State | Published - 1 Dec 2010 |

## Keywords

- Dual group
- Group of quasi-invariance
- Polish group
- Pontryagin duality theorem
- Quasi-convex group
- T-sequence

## ASJC Scopus subject areas

- Geometry and Topology