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.