Premium
On the equivalence of sets of equal Haar measures by countable decomposition
Author(s) -
Mycielski Jan,
Tomkowicz Grzegorz
Publication year - 2019
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12289
Subject(s) - mathematics , haar measure , countable set , abelian group , locally compact space , second countable space , conjecture , combinatorics , haar , invariant (physics) , discrete mathematics , equivalence (formal languages) , pure mathematics , compact group , lie group , artificial intelligence , computer science , wavelet , mathematical physics
Let G be a locally compact topological group with a left invariant Haar measure μ . In 1976, Chuaqui made the following conjecture: If μ is σ ‐finite andX , Y ⊆ Gare measurable sets with non‐empty interiors, thenμ ( X ) = μ ( Y )if and only if there exist two countable partitions of X and Y into Haar measurable setsA 1 , A 2 , …andB 1 , B 2 , … , respectively, and elementsg 1 , g 2 , … ∈ G , such thatg i A i = B ifori = 1 , 2 , … . This conjecture is still open even for the case of compact abelian groups, but we will prove it under the additional assumption: G is second countable and either totally disconnected or has at most countably many connected components.