Measures and forking
type: Item
instance of: scholarly article
title: Measures and forking
zbMATH Open document ID: 0633.03024
DOI: 10.1016/0168-0072(87)90069-8
author: H. Jerome Keisler
published in: Annals of Pure and Applied Logic
publication date: 1987-01-01T00:00:00Z
review text: The theory of forking for stable theories is generalized to a theory of measures for not necessarily stable theories. Let M be a very saturated model of countable T. Let \(A\subset M\) be small. We can identify a complete 1-type over A with a 0-1 valued measure on the A-definable subsets of M. Why not consider also finitely additive probability measures \(\mu\) on the A-definable subsets of M? Loeb measure shows that any such \(\mu\) extends to a unique countably additive measure \(\alpha\) on the A-Borel subsets of M and that for X open \(\alpha (X)=\sup \{\alpha (Y):\) Y A-definable, \(Y\subset X\}\). (By definition, an A-open set is an arbitrary union of A-definable sets, and the A-Borel sets are defined in the usual way). Such \(\alpha\) is called a measure over A. (Keisler actually considers measures over fragments in the sense of Harnik- Harrington.) A measure \(\beta\) over B does not fork over \(A\subset B\) if \(\beta [fk(B,A)]=0\) (where \(fk(B,A)=\{a\in M:\) for some finite \(\Delta\subset L\), \(R(tp(a/B),\Delta,\aleph_ 0)<R(tp(a/A),\Delta,\aleph_ 0)\})\). sbl(A) is by definition \(\{a\in M:\) \(tp(a/A)\) is stable, i.e. has finite \(R(-,\Delta,\aleph_ 0)\) for all finite \(\Delta\subset L\}\). \(usbl(A)=M-sbl(A)\). The measure \(\alpha\) over A is said to be smooth if for any 2 extensions \(\beta_ 1\), \(\beta_ 2\) of \(\alpha\) over \(B\supset A\), \(\beta_ 1\) and \(\beta_ 2\) agree on usbl(A). It is shown that for smooth measures and nonforking as defined above, the usual ``nonforking axioms'' are satisfied (existence, transitivity, countable basis, bounded number of nonforking extensions). (A complete type is smooth iff stable). It is moreover shown that T does not have the independence property just if every measure has an extension to a smooth measure. The notion of a flat extension is introduced: if N is big \(A\subset N\) and \(\beta\) is a measure over N, \(\beta\) is flat over A if \(\beta\) is invariant under A-automorphisms of N. This gives …
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 03H10
zbMATH DE Number: 4029524
zbMATH Keywords: unstable theories
zbMATH Keywords: forking
zbMATH Keywords: measures
zbMATH Keywords: saturated model
zbMATH Keywords: A-definable subsets
zbMATH Keywords: Loeb measure
zbMATH Keywords: A-Borel subsets
zbMATH Keywords: smooth measure
zbMATH Keywords: integration
zbMATH Keywords: nonforking product of measures
reviewed by: Anand Pillay
MaRDI profile type: Publication
full work available at URL: 0168-0072(87)90069-8
OpenAlex ID: W2049327281
cites work: Model theory.
cites work: Nonstandard Measure Theory and its Applications
cites work: Q5726943
cites work: Fundamentals of forking
cites work: An introduction to forking
cites work: Q3882458
cites work: Forking, normalization and canonical bases
cites work: Definable sets in ordered structures
cites work: Classification theory and the number of non-isomorphic models
cites work: Simple unstable theories
cites work: Foundations of infinitesimal stochastic analysis
description: scientific article; zbMATH DE number 4029524