Measures and forking
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 something new even in the case where T is stable and \(\beta\upharpoonright A\) is a complete type: \(\beta\) is the ``average'' of the nonforking extensions of \(\beta\upharpoonright A\) to N, which is itself only a complete type when \(\beta\upharpoonright A\) is stationary in the usual sense. If \(\beta\upharpoonright A\) is smooth then \(\beta\) (over \(N\supset A)\) is flat over A iff \(\beta\) is ``definable over A''; where \(\beta\) being definable over A means that for any \(\phi(x,\bar y)\in L\) the function \(f(\bar y)=\beta (\phi(x,\bar y))\) on \(\bar y\subset N\) is Borel over A. Finally using integration the nonforking product of measures \(\alpha\) and \(\beta\) is defined, giving as a consequence a ``forking symmetry'' result for smooth measures.
scientific article; zbMATH DE number 4029524
scientific article