Borel equivalence relations: structure and classification
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2008]
|
| Series: | University lecture series
Volume 44 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 240 Seiten graph. Darst. |
| ISBN: | 9780821844533 0821844539 |
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MARC
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| 020 | |a 0821844539 |9 0-8218-4453-9 | ||
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| 100 | 1 | |a Kanovej, Vladimir |e Verfasser |0 (DE-588)1051922119 |4 aut | |
| 245 | 1 | 0 | |a Borel equivalence relations |b structure and classification |c Vladimir Kanovei |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2008] | |
| 264 | 4 | |c © 2008 | |
| 300 | |a X, 240 Seiten |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a University lecture series |v Volume 44 | |
| 650 | 4 | |a Classes d'équivalence (Théorie des ensembles) | |
| 650 | 4 | |a Ensembles boréliens | |
| 650 | 7 | |a Ensembles boréliens |2 ram | |
| 650 | 4 | |a Relations d'équivalence (Théorie des ensembles) | |
| 650 | 7 | |a Relations d'équivalence (théorie des ensembles) |2 ram | |
| 650 | 0 | 7 | |a Äquivalenzrelation |0 (DE-588)4141500-0 |2 gnd |9 rswk-swf |
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Record in the Search Index
| _version_ | 1804138081986543616 |
|---|---|
| adam_text | Introduction
Chapter
1.
Descriptive
set theoretic background
1.1.
Polish spaces
1.2.
Point-sets.
Borei
sets
1.3.
Projective
sets
1.4.
Analytic, formulas
1.5.
Transformation of analytic formulas
1.6.
Effective hierarchies of pointsets
1.7.
Characterization of
Σ
sets
1.8.
Classifying functions
1.9.
Closure properties
Contents
Preface
ix
1
7
7
8
10
10
12
13
14
15
16
Chapter
2.
Some theorems of descriptive set theory
19
2.1.
Trees and ranks
19
2.2.
Trees and sets of the first
projective
level
22
2.3.
Reduction and separation
23
2.4.
Uniformization and
Kreisel
Selection
24
2.5.
Universal sets
27
2.6.
Good universal sets
29
2.7.
Reflection
30
2.8.
Enumeration of A sets
31
2.9.
Coding
Borei
sets
33
2.10.
Choquet property of £| and the Gandy-Harrington topology
34
2.11.
Sets with countable sections
36
2.12.
Applications for
Borei
sets
38
Chapter
3.
Borei
ideals
41
3.1.
Introduction to ideals
41
3.2.
Reducibility of ideals
41
3.3.
P-ideals and submeasures
43
3.4.
Polishable ideals
44
3.Õ.
Characterization of polishable ideals
45
3.6.
Simmiable and density ideals
47
3.7.
Operations on ideals and
Fréchet
ideals
49
3.8.
Some other ideals
49
Chapter
4.
Introduction to equivalence relations
51
4.1.
Some examples of
Borei
equivalence relations
51
vi
CONTENTS
4.2.
Operations on equivalence relations
52
4.3.
Orbit equivalence relations of group actions
54
4.4.
Some examples of orbit equivalence relations
55
4.5.
Probability measures
57
4.6.
Invariant and ergodic measures
58
Chapter
5.
Borei
reducibility of equivalence relations
63
5.1.
Borei
reducibility
63
5.2.
Injective
Borei
reducibility
—
embedding
64
5.3.
Borei,
continuous, and Baire meastirable reductions
65
5.4.
Additive reductions
66
5.5.
Diagram of
Borei
reducibility of key equivalence relations
67
5.6.
Reducibility and irreducibility on the diagram
68
5.7.
Dichotomy theorems
70
5.8.
Borei
ideals in the structure of
Borei
reducibility
71
Chapter
6.
Elementary results
73
6.1.
Equivalence relations E3 and T2
73
6.2.
Discretization and generation by ideals
74
6.3.
Summables
irreducible to density-0
76
6.4.
How to eliminate forcing
79
6.5.
The family £p
80
6.6.
£°°
:
maximal
Κσ
82
Chapter
7.
Introduction to countable equivalence relations
85
7.1.
Several types of equivalence relations
85
7.2.
Smooth and below
86
7.3.
Assembling countable equivalence relations
88
7.4.
Countable equivalence relations and group actions
89
7.5.
Non-hyperfinite countable equivalence relations
90
7.6.
A sufficient condition of essential countability
93
Chapter
8.
Hyperfmite equivalence relations
95
8.1.
Hyperfinite equivalence relations: The characterization theorem
95
8.2.
Proof of the characterization theorem
96
8.3.
Hyperfiniteness of tail equivalence relations
101
8.4.
Classification modulo
Borei
isomorphism
103
8.5.
Remarks on the classification theorem
104
8.6.
Which groups induce hyperfinite equivalence relations?
106
Chapter
9.
More on countable equivalence relations
107
9.1.
Amenable groups
108
9.2.
Amenable equivalence relations
109
9.3.
Hyperfiniteness and amenability 111
9.4.
Treeable equivalence relations
112
9.5.
Above treeable. Free
Borei
countable equivalence relations
113
Chapter
10.
The
1st
and
2nd
dichotomy theorems
119
10.1.
The
1st
dichotomy theorem
119
10.2.
Splitting system
121
CONTENTS
vii
10.3.
Structural
and chaotic domains
122
10.4. 2nd
dichotomy theorem
122
10.5.
Restricted product forcing
125
10.6.
Splitting system
126
10.7.
Construction of a splitting system
127
10.8.
The ideal of Eo -small sets
128
10.9.
A forcing notion associated with Eo
130
Chapter
11.
Ideal J* and the equivalence relation Ei
133
11.1.
Ideals below Jx
133
11.2.
Ei
:
hypersmoothness and non-countability
135
11.3. 3rd
dichotomy
136
11.4.
Case
1 138
11.5.
Case
2 138
11.6.
The construction
140
11.7.
A forcing notion associated with Ei
142
11.8.
Above Ei
143
Chapter
12.
Actions of the infinite symmetric group
147
12.1.
Infinite symmetric group S^ and isomorphisms
147
12.2.
Borei
invariant sets
148
12.3.
Equivalence relations classifiable by countable structures
149
12.4.
Reduction to countable graphs
150
12.5.
Reduction of
Borei
classifiability to
Ύξ
151
Chapter
13.
Turbulent group actions
155
13.1.
Local orbits and turbulence
155
13.2.
Shift actions of summable ideals are turbulent
156
13.3.
Ergodicity
157
13.4.
Generic reduction to
Τξ
158
13.5.
Ergodicity of turbulent actions w.r.t.
Ύζ
160
13.6.
Inductive step of countable power
161
13.7.
Inductive step of the Fubini product
163
13.8.
Other inductive steps
163
13.9.
Applications to the shift action of ideals
164
Chapter
14.
The ideal
Jř%
and the equivalence relation E3
167
14.1.
Continual assembling of equivalence relations
167
14.2.
The two cases
169
14.3.
Case
1 171
14.4.
Case
2 172
14.5.
Splitting system
173
14.6.
The embedding
174
14.7.
The construction of a splitting system: warmup
175
14.8.
The construction of a splitting system: the step
175
14.9.
A forcing notion associated with E3
178
Chapter
15.
Summable equivalence relations
181
15.1.
Classification of summable ideals and equivalence relations
181
15.2.
Grainy sets and the two cases
182
viii CONTENTS
15.3.
Case
1 183
15.4.
Case
2 185
15.5.
The construction of a splitting system
186
15.6.
A forcing notion associated with E2
187
Chapter
16.
Co-equalities
191
16.1.
Co-equalities: definition
191
16.2.
Some examples and simple results
192
16.3.
Co-equalities and additive reducibility
193
16.4.
A largest Co-equality
194
16.5.
Classification
195
16.6.
LV-equalities
197
16.7.
Non- a-compact
case
200
Chapter
17.
Pinned equivalence relations
203
17.1.
The definition of pinned equivalence relations
203
17.2.
T2 is not pinned
205
17.3.
Fubini product of pinned equivalence relations is pinned
205
17.4.
Complete left-invariant actions induce pinned relations
206
17.5.
All equivalence relations with
Σ°
classes are pinned
207
17.6.
Another family of pinned ideals
208
Chapter
18.
Reduction of
Borei
equivalence relations
to
Borei
ideals
211
18.1.
Trees
211
18.2.
Louveau-Rosendal transform
212
18.3.
Embedding and equivalence of normal trees
214
18.4.
Reduction to
Borei
ideals: first approach
216
18.5.
Reduction to
Borei
ideals: second approach
218
18.6.
Some questions
221
Appendix A. On Cohen and Gandy-Harrington forcing
оч-ег
countable models
223
A.I. Models of a fragment of ZFC
223
A.
2.
Coding uncountable sets in countable models
225
A.3. Forcing over countable models
225
A.4. Cohen forcing
227
A.5. Gandy-Harrington forcing
228
Bibliography
231
Index
235
|
| adam_txt |
Introduction
Chapter
1.
Descriptive
set theoretic background
1.1.
Polish spaces
1.2.
Point-sets.
Borei
sets
1.3.
Projective
sets
1.4.
Analytic, formulas
1.5.
Transformation of analytic formulas
1.6.
Effective hierarchies of pointsets
1.7.
Characterization of
Σ\'
sets
1.8.
Classifying functions
1.9.
Closure properties
Contents
Preface
ix
1
7
7
8
10
10
12
13
14
15
16
Chapter
2.
Some theorems of descriptive set theory
19
2.1.
Trees and ranks
19
2.2.
Trees and sets of the first
projective
level
22
2.3.
Reduction and separation
23
2.4.
Uniformization and
Kreisel
Selection
24
2.5.
Universal sets
27
2.6.
Good universal sets
29
2.7.
Reflection
30
2.8.
Enumeration of A\ sets
31
2.9.
Coding
Borei
sets
33
2.10.
Choquet property of £| and the Gandy-Harrington topology
34
2.11.
Sets with countable sections
36
2.12.
Applications for
Borei
sets
38
Chapter
3.
Borei
ideals
41
3.1.
Introduction to ideals
41
3.2.
Reducibility of ideals
41
3.3.
P-ideals and submeasures
43
3.4.
Polishable ideals
44
3.Õ.
Characterization of polishable ideals
45
3.6.
Simmiable and density ideals
47
3.7.
Operations on ideals and
Fréchet
ideals
49
3.8.
Some other ideals
49
Chapter
4.
Introduction to equivalence relations
51
4.1.
Some examples of
Borei
equivalence relations
51
vi
CONTENTS
4.2.
Operations on equivalence relations
52
4.3.
Orbit equivalence relations of group actions
54
4.4.
Some examples of orbit equivalence relations
55
4.5.
Probability measures
57
4.6.
Invariant and ergodic measures
58
Chapter
5.
Borei
reducibility of equivalence relations
63
5.1.
Borei
reducibility
63
5.2.
Injective
Borei
reducibility
—
embedding
64
5.3.
Borei,
continuous, and Baire meastirable reductions
65
5.4.
Additive reductions
66
5.5.
Diagram of
Borei
reducibility of key equivalence relations
67
5.6.
Reducibility and irreducibility on the diagram
68
5.7.
Dichotomy theorems
70
5.8.
Borei
ideals in the structure of
Borei
reducibility
71
Chapter
6.
"Elementary" results
73
6.1.
Equivalence relations E3 and T2
73
6.2.
Discretization and generation by ideals
74
6.3.
Summables
irreducible to density-0
76
6.4.
How to eliminate forcing
79
6.5.
The family £p
80
6.6.
£°°
:
maximal
Κσ
82
Chapter
7.
Introduction to countable equivalence relations
85
7.1.
Several types of equivalence relations
85
7.2.
Smooth and below
86
7.3.
Assembling countable equivalence relations
88
7.4.
Countable equivalence relations and group actions
89
7.5.
Non-hyperfinite countable equivalence relations
90
7.6.
A sufficient condition of essential countability
93
Chapter
8.
Hyperfmite equivalence relations
95
8.1.
Hyperfinite equivalence relations: The characterization theorem
95
8.2.
Proof of the characterization theorem
96
8.3.
Hyperfiniteness of tail equivalence relations
101
8.4.
Classification modulo
Borei
isomorphism
103
8.5.
Remarks on the classification theorem
104
8.6.
Which groups induce hyperfinite equivalence relations?
106
Chapter
9.
More on countable equivalence relations
107
9.1.
Amenable groups
108
9.2.
Amenable equivalence relations
109
9.3.
Hyperfiniteness and amenability 111
9.4.
Treeable equivalence relations
112
9.5.
Above treeable. Free
Borei
countable equivalence relations
113
Chapter
10.
The
1st
and
2nd
dichotomy theorems
119
10.1.
The
1st
dichotomy theorem
119
10.2.
Splitting system
121
CONTENTS
vii
10.3.
Structural
and chaotic domains
122
10.4. 2nd
dichotomy theorem
122
10.5.
Restricted product forcing
125
10.6.
Splitting system
126
10.7.
Construction of a splitting system
127
10.8.
The ideal of Eo -small sets
128
10.9.
A forcing notion associated with Eo
130
Chapter
11.
Ideal J*\ and the equivalence relation Ei
133
11.1.
Ideals below Jx
133
11.2.
Ei
:
hypersmoothness and non-countability
135
11.3. 3rd
dichotomy
136
11.4.
Case
1 138
11.5.
Case
2 138
11.6.
The construction
140
11.7.
A forcing notion associated with Ei
142
11.8.
Above Ei
143
Chapter
12.
Actions of the infinite symmetric group
147
12.1.
Infinite symmetric group S^ and isomorphisms
147
12.2.
Borei
invariant sets
148
12.3.
Equivalence relations classifiable by countable structures
149
12.4.
Reduction to countable graphs
150
12.5.
Reduction of
Borei
classifiability to
Ύξ
151
Chapter
13.
Turbulent group actions
155
13.1.
Local orbits and turbulence
155
13.2.
Shift actions of summable ideals are turbulent
156
13.3.
Ergodicity
157
13.4.
"Generic" reduction to
Τξ
158
13.5.
Ergodicity of turbulent actions w.r.t.
Ύζ
160
13.6.
Inductive step of countable power
161
13.7.
Inductive step of the Fubini product
163
13.8.
Other inductive steps
163
13.9.
Applications to the shift action of ideals
164
Chapter
14.
The ideal
Jř%
and the equivalence relation E3
167
14.1.
Continual assembling of equivalence relations
167
14.2.
The two cases
169
14.3.
Case
1 171
14.4.
Case
2 172
14.5.
Splitting system
173
14.6.
The embedding
174
14.7.
The construction of a splitting system: warmup
175
14.8.
The construction of a splitting system: the step
175
14.9.
A forcing notion associated with E3
178
Chapter
15.
Summable equivalence relations
181
15.1.
Classification of summable ideals and equivalence relations
181
15.2.
Grainy sets and the two cases
182
viii CONTENTS
15.3.
Case
1 183
15.4.
Case
2 185
15.5.
The construction of a splitting system
186
15.6.
A forcing notion associated with E2
187
Chapter
16.
Co-equalities
191
16.1.
Co-equalities: definition
191
16.2.
Some examples and simple results
192
16.3.
Co-equalities and additive reducibility
193
16.4.
A largest Co-equality
194
16.5.
Classification
195
16.6.
LV-equalities
197
16.7.
Non- a-compact
case
200
Chapter
17.
Pinned equivalence relations
203
17.1.
The definition of pinned equivalence relations
203
17.2.
T2 is not pinned
205
17.3.
Fubini product of pinned equivalence relations is pinned
205
17.4.
Complete left-invariant actions induce pinned relations
206
17.5.
All equivalence relations with
Σ°
classes are pinned
207
17.6.
Another family of pinned ideals
208
Chapter
18.
Reduction of
Borei
equivalence relations
to
Borei
ideals
211
18.1.
Trees
211
18.2.
Louveau-Rosendal transform
212
18.3.
Embedding and equivalence of normal trees
214
18.4.
Reduction to
Borei
ideals: first approach
216
18.5.
Reduction to
Borei
ideals: second approach
218
18.6.
Some questions
221
Appendix A. On Cohen and Gandy-Harrington forcing
оч-ег
countable models
223
A.I. Models of a fragment of ZFC
223
A.
2.
Coding uncountable sets in countable models
225
A.3. Forcing over countable models
225
A.4. Cohen forcing
227
A.5. Gandy-Harrington forcing
228
Bibliography
231
Index
235 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Kanovej, Vladimir |
| author_GND | (DE-588)1051922119 |
| author_facet | Kanovej, Vladimir |
| author_role | aut |
| author_sort | Kanovej, Vladimir |
| author_variant | v k vk |
| building | Verbundindex |
| bvnumber | BV035109863 |
| callnumber-first | Q - Science |
| callnumber-label | QA248 |
| callnumber-raw | QA248 |
| callnumber-search | QA248 |
| callnumber-sort | QA 3248 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 430 |
| ctrlnum | (OCoLC)470947948 (DE-599)HBZHT015479204 |
| dewey-full | 511.322 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.322 |
| dewey-search | 511.322 |
| dewey-sort | 3511.322 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV035109863 |
| illustrated | Illustrated |
| index_date | 2024-07-02T22:17:05Z |
| indexdate | 2024-07-09T21:22:31Z |
| institution | BVB |
| isbn | 9780821844533 0821844539 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016777706 |
| oclc_num | 470947948 |
| open_access_boolean | |
| owner | DE-824 DE-355 DE-BY-UBR DE-83 DE-11 DE-188 DE-91G DE-BY-TUM |
| owner_facet | DE-824 DE-355 DE-BY-UBR DE-83 DE-11 DE-188 DE-91G DE-BY-TUM |
| physical | X, 240 Seiten graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | University lecture series |
| series2 | University lecture series |
| spelling | Kanovej, Vladimir Verfasser (DE-588)1051922119 aut Borel equivalence relations structure and classification Vladimir Kanovei Providence, Rhode Island American Mathematical Society [2008] © 2008 X, 240 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier University lecture series Volume 44 Classes d'équivalence (Théorie des ensembles) Ensembles boréliens Ensembles boréliens ram Relations d'équivalence (Théorie des ensembles) Relations d'équivalence (théorie des ensembles) ram Äquivalenzrelation (DE-588)4141500-0 gnd rswk-swf Borel-Menge (DE-588)4146323-7 gnd rswk-swf Borel-Menge (DE-588)4146323-7 s Äquivalenzrelation (DE-588)4141500-0 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-2188-5 University lecture series Volume 44 (DE-604)BV004153846 44 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016777706&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kanovej, Vladimir Borel equivalence relations structure and classification University lecture series Classes d'équivalence (Théorie des ensembles) Ensembles boréliens Ensembles boréliens ram Relations d'équivalence (Théorie des ensembles) Relations d'équivalence (théorie des ensembles) ram Äquivalenzrelation (DE-588)4141500-0 gnd Borel-Menge (DE-588)4146323-7 gnd |
| subject_GND | (DE-588)4141500-0 (DE-588)4146323-7 |
| title | Borel equivalence relations structure and classification |
| title_auth | Borel equivalence relations structure and classification |
| title_exact_search | Borel equivalence relations structure and classification |
| title_exact_search_txtP | Borel equivalence relations structure and classification |
| title_full | Borel equivalence relations structure and classification Vladimir Kanovei |
| title_fullStr | Borel equivalence relations structure and classification Vladimir Kanovei |
| title_full_unstemmed | Borel equivalence relations structure and classification Vladimir Kanovei |
| title_short | Borel equivalence relations |
| title_sort | borel equivalence relations structure and classification |
| title_sub | structure and classification |
| topic | Classes d'équivalence (Théorie des ensembles) Ensembles boréliens Ensembles boréliens ram Relations d'équivalence (Théorie des ensembles) Relations d'équivalence (théorie des ensembles) ram Äquivalenzrelation (DE-588)4141500-0 gnd Borel-Menge (DE-588)4146323-7 gnd |
| topic_facet | Classes d'équivalence (Théorie des ensembles) Ensembles boréliens Relations d'équivalence (Théorie des ensembles) Relations d'équivalence (théorie des ensembles) Äquivalenzrelation Borel-Menge |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016777706&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV004153846 |
| work_keys_str_mv | AT kanovejvladimir borelequivalencerelationsstructureandclassification |