scholarly journals A nonseparable invariant extension of Lebesgue measure – A generalized and abstract approach

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sanjib Basu ◽  
Debasish Sen
Keyword(s):  
Lebesgue Measure ◽  
Set Theory ◽  
Transformation Groups ◽  
Invariant Extension ◽  
Abstract Approach ◽  
Combinatorial Set ◽  
Independent Families ◽  
Combinatorial Set Theory

Abstract In this paper, we use some methods of combinatorial set theory, in particular, the ones related to the construction of independent families of sets and also some modified version of the notion of small sets originally introduced by Riečan and Neubrunn, to give an abstract and generalized formulation of a remarkable theorem of Kakutani and Oxtoby related to a nonseparable invariant extension of the Lebesgue measure in spaces with transformation groups.

Notes on combinatorial set theory

1973 ◽  
Vol 14 (3) ◽  
pp. 262-277 ◽  
Author(s):  
Saharon Shelah
Keyword(s):  
Set Theory ◽  
Combinatorial Set ◽  
Combinatorial Set Theory

scholarly journals Morasses and the Lévy-collapse

1987 ◽  
Vol 52 (1) ◽  
pp. 111-115 ◽  
Author(s):  
P. Komjáth
Keyword(s):  
Set Theory ◽  
Present Author ◽  
Large Cardinal ◽  
Inaccessible Cardinal ◽  
Mahlo Cardinal ◽  
Counter Example ◽  
Survey Papers ◽  
Combinatorial Set ◽  
Combinatorial Set Theory

For several old problems in combinatorial set theory A. Hajnal and the present author [2] showed that on collapsing a sufficiently Mahlo cardinal to ω1 by the Lévy-collapse one gets a model where these problems are solved in the “counter-example” direction. The authors of [2] have speculated that the theorems of that paper should hold in L, and this, in fact, was shown for some of the results by Todorčević and Velleman [7,8]. The observation that collapsing a large cardinal to ω1 may give rise to L-like constructions is not new. As it was shown long ago by Silver and Rowbottom, there is a Kurepa-tree if a strongly inaccessible cardinal is Lévy-collapsed to ω1. In [5] it is proved that even Silver's W holds in that model. Here we show that even a quagmire exists there, but not necessarily a morass. To be more exact, we show that if κ < λ are the first two strongly inaccessible cardinals, first λ is Lévy-collapsed to κ+, and then κ is Lévy-collapsed to then there is no ω1-morass with built-in diamond in the resulting model (GCH is assumed). If λ is Mahlo, there is not even a morass.Our notations are standard. For excellent survey papers on morass-like principles and their uses in combinatorial set theory see [4,5,6].

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