Finding generically stable measures

Pierre Simon

January 25, 2011

Abstract This work builds on [6] and [7] where Keisler measures over NIP theories are studied. We discuss two constructions for obtaining generically stable measures in this context. First, we show how to symmetrize an arbitrary invariant measure to obtain a generically stable one from it. Next, we show that suitable sigma-additive probability measures give rise to generically stable Keisler measures. Also included is a proof that generically stable measures over o-minimal theories and the p-adics are smooth.

Introduction

A major theme in the study of NIP theories is the investigation of stable phenomena in them. Generically stable types are defined in [6]: they are types that behave, as far as non-forking extensions are concerned, like stable types. This notion was generalized to Keisler measures in [7]. Keisler measures (introduced first in [9] and studied again with a different approach in [5], [6] and [7]) are regular Borel probability measures on the space of types. Equivalently, they are finitely additive probability measures on the boolean algebra of definable sets. They can also be considered as types in continuous logic (see [3]). Some basic facts about them are recalled in the first section.

After Keisler’s seminal work [9], measures were introduced again for the study of NIP theories in Hrushovski, Peterzil and Pillay’s paper [5], focussing mainly on invariant measures of groups. The notion of fsg group is introduced. Measures are studied more at depth in [6] in order to prove compact domination for definably compact groups in o-minimal theories. Again, applications use only invariant measures on groups, but a general study is initiated. Also, the notion of generically stable type is defined and the question is asked of a generalization to measures. This is done in [7] by Hrushovski, Pillay and the author where generically stable measures are defined. A number of equivalent properties are given.

The following examples of generically stable measures are known:

- a generically stable type: this is the motivating example,

- the (translation) invariant measure of an fsg group,

- the A-invariant measure extending an fsg type over A ([7] section 4),

- the average measure of an indiscernible segment ([7], 3.7),

- the Keisler measure induced by a σ-additive measure on the standard model, ℝ or ℚ_{p} ([7], section 6). In fact, those measures are proven to be smooth, a stronger property.

In this paper, we generalize the last two constructions in this list. First, we show how to symmetrize any measure, or equivalently to average an indiscernible segment of measures. Second, we show that any σ-additive measure induces a generically stable measure, under the assumption that externally definable sets in two variables are measurable (so that Fubini applies). Smoothness of the induced measure is not true in general, and to recover the full result of [7] we also include a proof that all generically stable measures in ℝ or ℚ_{p} are smooth. In order to prove those results, we first establish in Section 2 a criterion to recognize a product measure μ_{x} ⊗ λ_{y} when μ_{x} happens to be finitely satisfiable over some small model. Our strategy for proving that a given measure μ is generically stable will then be to construct a symmetric measure η_{x_{1}x_{2}...} in ω variables, and show using this criterion that it is the Morley sequence of μ.

We describe the canonical example of a generically stable measure we have in mind. Consider the following theory: the signature is {≤, E}. The reduct to {≤} is a dense linear order, and the theory says that E is an equivalence relation with infinitely many classes, each of which is dense with respect to ≤. This theory has no generically stable type over the main sort (because every type falls in some cut of the linear order). However, one can build a generically stable measure over the main sort by averaging types that fall in different cuts. More precisely, assume for example that we work over a model (M, ≤, E) and we have an increasing embedding f : ((0, 1), ≤) ) (M, ≤), where (0, 1) denotes the standard unit open interval. Let λ_{0} denote the standard Lebesgue measure on (0, 1). Define a Keisler measure μ on M by μ(a ≤ x) = λ_{0}(f^{−1}([a, + ))) and μ(aEx) = 0 for all a ∈ M. This measure is generically stable. It lifts the generically stable generic type of the imaginary sort M/E. We will see that this phenomenon is general: we can always lift generically stable types in an imaginary sort to generically stable measures on the main sort (see the proof of Lemma 4.5).

Of course this example is rather special in that the generic type of M/E is stable and not just generically stable. Here is an example where this is not the case. Start with the structure with universe ℚ and with language {P_{n}(x, y) : n < ω} where P_{n}(x, y) holds if and only if |x − y| < n. Then there is a generically stable type p at infinity. Now expand that structure with a generic linear order < (such that every infinite definable set in the reduct is dense-co-dense with respect to <). Then the type p lifts in many ways to a generically stable measure, by taking the union of it with some Lebesgue measure on <. We thus obtain a non-smooth generically stable measure, but there are no generically stable types in the expanded theory, even in M^{eq}.

We will use standard notation. We will work with a complete first order theory T in some language L; T is assumed to be NIP throughout the paper. For simplicity, we assume that T is one-sorted and work in T^{eq}. We have a monster model M̄; M, N,... will denote small submodels of M̄, and A, B, C... small parameter sets. We will not distinguish between points and tuples; they will be named by a, b, c... and x, y, z... will designate variables of finite or infinite tuples. The notation L_{x}(A) denotes the set of formulas with parameters in A and free variable x.

The space of types over A in variable x is designated by S_{x}(A). It is equipped with the usual compact topology and the associated σ-algebra of Borel subsets. By “X is Borel over A”, we mean that it is a Borel subset of some S_{x}(A). We write a ≡_{M} b for tp(a/M) = tp(b/M).

By a global type or measure, we mean a type or a measure over M̄.

1 Preliminaries

We recall some basic facts about Keisler measures. We will be brief, and the reader is refered to [6] and [7] for more details.

We make a blanket assumption that T is NIP.

Basic definitions

A Keisler measure (or simply a measure) over A in variable x is a finitely additive probability measure on the boolean algebra L_{x}(A) of formulas with free variable x and parameters in A. As in section 4 of [6], such a measure extends uniquely to a regular Borel probability (σ-additive) measure on the type space S_{x}(A). (“Regular” means that the measure of any Borel set X is the infimum of the measures of open sets O such that X ⊆ O. Furthermore, in our situation, working on a totally disconnected space, the measure of O is itself the supremum of measures of clopen sets inside it.)

Conversely, given a regular Borel measure on S_{x}(A), its restriction to the clopen sets gives a Keisler measure.

We will denote by ℳ_{x}(A) the space of Keisler measures over A and often write μ_{x} ∈ ℳ(A) for μ ∈ ℳ_{x}(A), keeping track of the variable in the name of the measure. We can consider ℳ_{x}(A) as a subset of [0, 1]^{L_{x}(A)}. It inherits the product topology making it a compact Hausdorff space.

Lemma 1.1. Let Ω ⊆ L_{x}(A) be a set of formulas closed under intersection, union and complement and containing ⊤. Let μ_{0} be a finitely additive measure on Ω with values in [0, 1] such that μ_{0}(⊤) = 1. Then μ extends to a Keisler measure over A.

Proof. By compactness in the space [0, 1]^{L_{x}(A)}, it is enough to show that given ψ_{1}(x), .., ψ_{n}(x) formulas in L_{x}(A), there is a function f = ⟨ψ_{1}, .., ψ_{n}⟩ ) [0, 1] finitely additive and compatible with μ_{0} (where ⟨B⟩ denotes the boolean algebra generated by B). We may assume that ψ_{1}, .., ψ_{n} are the atoms of the boolean algebra B that they generate.

The elements of Ω in B form a sub-boolean algebra. Let ϕ_{1}, .., ϕ_{m} be its atoms. We have say:

ϕ_{1} = ψ_{1} ∨ ... ∨ ψ_{i_{1}} ϕ_{2} = ψ_{i_{2}} ∨ ... ∨ ψ_{i_{3}} etc.

Then any finitely additive f satisfying f(ψ_{1}) + ... + f(ψ_{i_{1}} ) = μ_{0}(ϕ_{1}) etc. will do.

Here are some basic definitions:

Definition 1.2. Let M ≺ N, with N |M|^{+}-saturated and let μ_{x} ∈ ℳ(N),

• μ is finitely satisfiable in M if for every ϕ ∈ L_{x}(N) such that μ(ϕ) > 0, there is a ∈ M such that N ⊧ ϕ(a).

• μ is M-invariant if for every ϕ(x; y) ∈ L, and b ≡_{M} b′, μ(ϕ(x; b)) = μ(ϕ(x; b′)).

• μ is definable over M if it is M-invariant and for every ϕ(x; y) ∈ L, and r ∈ [0, 1], the set {p ∈ S_{y}(M) : μ(ϕ(x; b)) ≤ r for any b ∈ N, b ⊧ p} is a closed set of S_{y}(M).

• μ is Borel-definable over M if the above set is a Borel set of S_{y}(M).

Proposition 1.3 ([6] 4.9). If μ ∈ ℳ(N) is M-invariant (N is |M|^{+}-saturated), then it is Borel definable over M.

The support of μ ∈ ℳ(N) is the set of types p ∈ S(N) weakly random for μ, i.e., such that p ⊢ ¬ϕ(x) for every ϕ(x) ∈ L(N) such that μ(ϕ(x)) = 0. We will denote the support of μ by S(μ); it is a closed set of S(N). Note that if μ is finitely satisfiable in M then every type in S(μ) is also finitely satisfiable in M.

The next proposition says that in NIP theories, measures can be well approximated by averages of types. We use the notation Fr(X; a_{1}, .., a_{n}) which stands for \frac{1}{n}|\{i : a_{i} ∈ X\}|.

Proposition 1.4 ([7] 2.8). Let μ_{x} be any measure over M. Let ϕ(x, y) ∈ L, ϵ > 0, and let X_{1}, .., X_{k} be Borel sets over M. Then for all large enough n there are a_{1}, .., a_{n} such that for all r = 1, .., k and all b ∈ M, μ(X_{r} ∩ ϕ(x, b)) is within ϵ of Fr(X_{r} ∩ ϕ(x, b); a_{1}, .., a_{n}).

Taking X_{1} = S(μ) in the above proposition, we see that we can always assume that tp(a_{i}/M) ∈ S(μ) for every index i.

Invariant extensions

As in the case of types, the study of Keisler measures differs from measure theory in that the space in two dimensions is not the product of the one dimensional spaces, and there are in general different ways to amalgamate two measures in one variable to form a measure in two variables. We recall here the basic construction of invariant extensions.

Let M ≺ N, N being |M|^{+}-saturated, and μ_{x} ∈ ℳ(N) be M-invariant. If λ_{y} ∈ ℳ(N) is any measure, then we can define the invariant extension of μ_{x} over λ_{y}, denoted μ_{x} ⊗ λ_{y}. It is a measure in the two variables x, y defined the following way. Let ϕ(x, y) ∈ L(N). Take a small model P ≺ N containing M and the parameters of ϕ. Define μ_{x} ⊗ λ_{y}(ϕ(x, y)) = ∫ f(y)dλ_{y}, the integral ranging over S_{y}(P) where f(p) = μ(ϕ(x, b)) for b ∈ N, b ⊧ p (this function is Borel-measurable by Borel-definability of μ). It is easy to check that this does not depend on the choice of P.

If λ_{y} is also invariant, we can also form the product λ_{y} ⊗ μ_{x}. In general it will not be the case that λ_{y} ⊗ μ_{x} = μ_{x} ⊗ λ_{y}.

If μ_{x} is a global M-invariant measure, we define by induction: μ_{x_{1}...x_{n}}^{(n)} by μ_{x_{1}}^{(1)} = μ and μ_{x_{1}...x_{n+1}}^{n+1} = μ_{x_{n+1}} ⊗ μ_{x_{1}...x_{n}}^{(n)} . We let μ_{x_{1}x_{2}...}^{(ω)} be the union and call it the Morley sequence of μ. It is an indiscernible sequence in the following sense.

Definition 1.5. A measure μ_{x_{1}x_{2}...} is indiscernible over A if for every ϕ(x_{1}, .., x_{n}) ∈ L(A) and indices i_{1} < ... < i_{n}, we have

μ(ϕ(x_{1}, .., x_{n})) = μ(ϕ(x_{i_{1}} , .., x_{i_{n}} )).

We define in the same way μ_{x_{1}x_{2}...} to be totally indiscernible by removing in the above definition the assumption that the indices i_{1}, .., i_{n} are ordered.

We will need the following result from [2] (see also [7], 2.10).

Proposition 1.6. If μ_{x_{1},x_{2},...} ∈ ℳ(M) is indiscernible and ω_{y,x_{1},x_{2},...} extends μ, then for every formula ϕ(x, y) ∈ L(M), lim_{i)ω} ω(ϕ(x_{i}, y)) exists. Equivalently, for any ϕ(x, y) and ϵ > 0, there is N such that for any measure ω_{y,x_{1},x_{2},...} as above, we have |ω(ϕ(x_{i}, y)) − ω(ϕ(x_{i+1}, y))| ≥ ϵ for at most N values of i.

Generically stable measures

This paper is concerned with building generically stable measures. They are measures that behave very much like types in a stable theory (at least as far as non-forking extensions are concerned). Generically stable types were defined in [6] and this definition was naturally extended to measures in [7]. We recall here some equivalent definitions (a few others are given in [7], Theorem 3.3).

Theorem 1.7 (Generically stable measure). Let μ_{x} be a global M-invariant measure. Then the following are equivalent:

1. μ is both definable and finitely satisfiable (necessarily over M),

2. μ^{(ω)} |_{M} is totally indiscernible, 3. for any global M-invariant Keisler measure λ_{y}, μ_{x} ⊗ λ_{y} = λ_{y} ⊗ μ_{x},

4. μ commutes with itself: μ_{x} ⊗ μ_{y} = μ_{y} ⊗ μ_{x}.

If μ satisfies one of those properties, we say it is generically stable.

Let μ_{x} be a measure, and f a definable map whose range is the sort of the variable x. Then one can consider the push-forward measure λ_{y} = f_{∗}(μ) defined by λ_{y}(ϕ(y)) = μ_{x}(ϕ(f(x))). This is again a Keisler measure. If μ is definable over M (resp. finitely satisfiable in M) and f is M-definable, then f_{∗}(μ) is again definable over M (resp. finitely satisfiable in M). In particular, if μ is a global generically stable measure, then f_{∗}(μ) is also generically stable.

Smooth measures

Definition 1.8 (Smooth). A measure μ ∈ ℳ(N) is smooth if μ has a unique global extension. If M ⊂ N, we will say that μ is smooth over M is μ|_{M} is smooth.

The following important properties are proved in [7].

Proposition 1.9 ([7], 2.3). Let μ_{x} be smooth over M and let ϕ(x, y) ∈ L and ϵ > 0. Then there are formulas ν_{i}^{1}(x), ν_{i}^{2}(x) and ψ_{i}(y) for i = 1, .., n in L(M) such that:

1. The formulas ψ_{i}(y) form a partition of the y-space,

2. for all i and b ∈ M̄, if ψ_{i}(b) holds, then M̄ ⊧ ν_{i}^{1}(x) ) ϕ(x, b) ) ν_{i}^{2}(x),

3. for each i, μ(ν_{i}^{2}(x)) − μ(ν_{i}^{1}(x)) < ϵ.

Note that conversely, if the conclusion holds for all ϕ(x, y) and ϵ, then μ is smooth.

Corollary 1.10. If μ is smooth over N, then:

1. there is M ≺ N of size |T| such that μ is smooth over M,

2. μ is definable and finitely satisfiable in N (in particular μ is generically stable),

3. if λ_{y} is a measure over N, then there is a unique separated amalgam ω_{xy} of μ_{x} and λ_{y} (see Definition 2.1).

Will also need the following fact (initially from [9]), that we consider as a way to realize measures.

Lemma 1.11 ([7], 2.2). Let μ_{x} be a measure over M. Then there is an extension M ≺ N and a measure μ′_{x} over N extending μ such that μ′ is smooth.

Some additional facts about smooth measures can be found in Section 4.

2 Amalgams

We fix throughout this section the following objects: M ≺ N two models, with N being |M|^{+}-saturated, and μ_{x}, λ_{y} two measures over N. An amalgam of μ_{x} and λ_{y} is simply a measure ω_{xy} extending μ_{x} ∪ λ_{y}. We are interested in the characterization of different possible amalgams, especially ‘free’ amalgams.

The most basic property of an amalgam is independence in the sense of probability theory, which we call separation.

Definition 2.1 (Separated). The amalgam ω_{xy} is separated if

ω(ϕ(x) ∧ ψ(y)) = μ(ϕ(x)).λ(ψ(y))

for all ϕ(x), ψ(y) ∈ L(N).

Note that if μ is M-invariant, then μ ⊗ λ is separated. Also if μ or λ is a type, then every amalgam is separated.

We now go on to define when the amalgam ω_{xy} is a finitely satisfiable extension of μ. A natural attempt would be to ask for example that ω(θ(x, y)) ≤ sup_{a∈M} λ(θ(a, y)). However, this seems to be too weak, and we will ask for something stronger, allowing to ‘localize’ on some clopen ϕ(x).

Definition 2.2 (f.s. extension). The amalgam ω_{xy} is an f.s. extension in M of μ over λ if the following holds for all θ(x, y), ϕ(x) ∈ L(N):

ω(θ(x, y) ∧ ϕ(x)) ≤ μ(ϕ(x)). sup_{a∈ϕ(M)} λ(θ(a, y)).

First some basic observations. If ω is an f.s. extension in M of μ, then it is a separated amalgam (apply the definition with θ(x, y) = ψ(y) and then θ(x, y) = ¬ψ(y)). Also, the existence of such an amalgam implies that μ itself is finitely satisfiable in M (hence M-invariant).

Assume λ is a type realized by some a. We can view ω_{xy} as ω′_{x} ∈ ℳ(Na). Then ω_{xy} is an f.s. extension in M of μ if and only if ω′_{x} is finitely satisfiable in M.

In the following propositions, we use the notation Av(X_{i} : i = 1, .., n) to mean \frac{1}{n}|\{i : X_{i} holds \}|.

Proposition 2.3. Assume that μ is finitely satisfiable in M, then ω_{xy} = μ_{x} ⊗ λ_{y} is an f.s. extension (in M) of μ over λ.

Proof. We first assume that μ_{x} = p_{x} is a type. Let θ(x, y) ∈ L(N) and ϕ(x) ∈ L(N) such that p ⊢ ϕ(x), and take a small model P ⊂ N containing M and the parameters of θ.

Then ω(θ(x, y)) = λ(B) where B ⊆ S(N) is the (P-invariant) Borel subset B = {q : p ⊢ θ(x, b) for some (any) b ⊧ q}. Let ϵ > 0. By NIP (see 1.4), there are points b_{1}, .., b_{n} ∈ N such that:

|λ(B) − Av(b_{i} ∈ B : i = 1..n)| ≤ ϵ,
∀a ∈ P, |λ(θ(a, y)) − Av(θ(a, b_{i}) : i = 1...n)| ≤ ϵ.

As p is finitely satisfiable in M, there is a_{0} ∈ ϕ(M) such that for every i ∈ {1, .., n}:

p ⊢ θ(x, b_{i}) ⌊ N ⊧ θ(a_{0}, b_{i}).

Now p ⊢ θ(x, b_{i}) {} b_{i} ∈ B so:

λ(B) ≈ Av(b_{i} ∈ B)
= Av(p ⊢ θ(x, b_{i}))
= Av(θ(a_{0}, b_{i}))
≈ λ(θ(a_{0}, y)).

(Where x ≈ y means |x − y| ≤ ϵ.)

So |λ(B) − λ(θ(a_{0}, y))| ≤ 2ϵ. As this is true for all ϵ > 0, and remembering λ(B) = ω(θ(x, y)), we deduce ω(θ(x, y)) ≤ sup_{a∈ϕ(M)} λ(ϕ(a, y)). This finishes the proof in the cas μ = p.

We now consider the general case. Let as above θ(x, y), ϕ(x) ∈ L(P). Let ϵ > 0. By Proposition 1.4 we can find p_{1}, …, p_{n} ∈ S(μ) ⊂ S(N) such that:

∀b ∈ N, |μ(θ(x, b)) − Av(p_{i} ⊢ θ(x, b))| ≤ ϵ

and:

|μ(ϕ(x)) − Av(p_{i} ⊢ ϕ(x))| ≤ ϵ.

Note that p_{1}, …, p_{n} are finitely satisfiable in M. Let, for b ∈ M, f(b) = μ(θ(x, b) ∧ ϕ(x)) and f_{n}(b) = \frac{1}{n}.Card\{k : p_{k} ⊢ θ(x, b) ∧ ϕ(x)\}. Let m be the number of indices i for which p_{i} ⊢ ϕ(x).

Then,

ω(θ(x, y) ∧ ϕ(x)) = ∫ f(y)dλ_{y}
≤ ∫ f_{n}(y)dλ_{y} + ϵ
≤ \frac{m}{n} sup_{k}λ(\{b | p_{k} ⊢ θ(x, b) ∧ ϕ(x)\}) + ϵ
≤ \frac{m}{n} sup_{a∈ϕ(M)} λ(θ(a, y)) + ϵ
≤ μ(ϕ(x)) sup_{a∈ϕ(M)} λ(θ(a, y)) + 2ϵ.

(We use the first part of the proof to go from line 3 to 4).

As ϵ was arbitrary, we are done.

We now show that if μ is finitely satisfiable in M, then the invariant extension is the only f.s. extension of μ over λ.

Proposition 2.4. Assume that μ is finitely satisfiable in M and ω_{xy} is an f.s. extension in M of μ_{x}, then ω_{xy} = μ_{x} ⊗ λ_{y}.

Proof. Let θ(x, y) ∈ L(N) and let ϵ > 0. Let P ⊂ N be a small model containing M and the parameters of θ. By 1.4 we can find b_{1}, …, b_{n} ∈ N such that

|λ(θ(a, y)) − Av(θ(a, b_{i}))| ≤ ϵ, for all a ∈ P.

For all k ∈ {0, …, n}, let B_{k}(x) be the formula saying: “there are exactly k values of i for which θ(x, b_{i}) is true”.

Then:

ω(θ(x, y)) = ∑_{k} ω(θ(x, y) ∧ B_{k}(x))

≤ ∑_{k} μ(B_{k}(x)). sup_{a∈B_{k} (M)} λ(θ(a, y))

≤ ∑_{k} \frac{k}{n}μ(B_{k}(x)) + ϵ.

(We use finite satisfiability of μ on line 2).

Similarly,

ω(¬θ(x, y)) ≤ ∑_{k} (1 − \frac{k}{n})μ(B_{k}(x)) + ϵ,

and therefore

|ω(θ(x, y)) − ∑_{k} \frac{k}{n}μ(B_{k}(x))| ≤ ϵ.

Letting ϵ ) 0, we see that ω(θ(x, y)) is uniquely determined by μ and λ and the fact that ω is an f.s. extension of μ. By the previous proposition, we have ω_{xy}(θ(x, y)) = μ_{x} ⊗ λ_{y}(θ(x, y)).

3 Symmetrizations

The following construction already appears in [7]. Let I = ⟨a_{t} : t ∈ [0, 1]⟩ be an indiscernible sequence indexed by [0, 1] (we will call this an indiscernible segment). If ϕ(x, y) is a formula and b ∈ M̄, then NIP implies that ⌊ϕ(x, b)⌋ := {t ∈ [0, 1] :⊧ ϕ(a_{t}, b)} is a finite union of intervals and points. Let m denote the Lebesgue measure on [0, 1]. Then we can define a global measure μ = Av(I) by μ(ϕ(x, b)) = m(⌊ϕ(x, b)⌋). It is called the average measure of I.

Lemma 3.1. For any indiscernible segment I ⊂ M, the average measure μ = Av(I) is generically stable over M. Furthermore, for any n and ϕ(x_{1}, .., x_{n}) ∈ L(M̄), μ^{(n)}(θ(x_{1}, .., x_{n})) = ∫_{t_{1}∈[0,1]} .. ∫_{t_{n} ∈[0,1]} θ(x_{a_{1}} , .., x_{a_{n}} )dt_{1}...dt_{n}.

Proof. First notice that μ is finitely satisfiable in M by construction. It is easy to check directly from the definition that the formula given for μ^{(n)} is valid. From this, it follows that μ^{(n)} is symmetric for all n, therefore μ is generically stable.

If p ∈ S(M) is any type. Let p̃ be an M-invariant global extension of p (for example, a coheir). Let I be a Morley segment of p̃ (i.e., a Morley sequence indexed by [0, 1]). Then Av(I) is a generically stable measure and extends p. Note that if p is already generically stable, then μ = p. In general, by the previous lemma, the Morley sequence of μ is a symmetrization of the Morley sequence of p, i.e. μ^{(n)}|_{M} is the average over all permutation of variables of p^{(n)}|_{M}:

μ^{(n)}(θ(x_{1}, .., x_{n})) = \frac{1}{n!} ∑_{σ∈𝔖_{n}} p^{(n)}(θ(x_{σ(1)}, .., x_{σ(n)} )),

for θ(x_{1}, .., x_{n}) ∈ L(M). For this reason, we will call μ a symmetrization of p.

Our aim now is to define the same construction starting with a measure instead of a type p. So let μ ∈ ℳ(M) be any measure. Take μ̃ a global M-invariant extension of μ. We have defined the Morley sequence μ̃_{x_{1}x_{2}...}^{(ω)} of μ̃ and we can analogously define its Morley segment μ̃([0,1]) ; x̄ where x̄ stands for ⟨x_{t} : t ∈ [0, 1]⟩. Now let νx̄, be a smooth extension of μ̃^{([0,1])} (this is the analogue of taking a realization of a Morley segment of p). For any b ∈ M̄, consider the function f_{ϕ(x,b)} : t ) ν(ϕ(x_{t}, b)). By Proposition 1.6, this function has only countably many points of discontinuity. It is therefore integrable on [0, 1].

We can consider the integral μ_{Σ}(ϕ(x, b)) = ∫_{t∈[0,1]} ν(ϕ(x_{t}, b))dt. This defines a global Keisler measure extending μ.

Proposition 3.2. The measure μ_{Σ} constructed above is generically stable. Furthermore, for every n, and θ(x_{1}, .., x_{n}) ∈ L(M),

μ_{Σ}^{(n)}(θ(x_{1}, .., x_{n})) = \frac{1}{n!} ∑_{σ∈𝔖_{n}} μ̃^{(ω)} (θ(x_{σ(1)}, .., x_{σ(n)} )).

Proof. Let P be a small model containing M over which ν is smooth. Then ν is finitely satisfiable in P, and it follows that μ_{Σ} is also finitely satisfiable in P. Define an n-ary global measure λ_{x_{1}..x_{n}}^{n} by

λ^{n}(θ(x_{1}, .., x_{n})) = ∫_{t_{1}∈[0,1]} .. ∫_{t_{n} ∈[0,1]} ν(θ(x_{t_{1}} , .., x_{t_{n}} ))dt_{1}..dt_{n},

for any formula θ(x_{1}, .., x_{n}) ∈ L(M̄).

We will show by induction that λ^{n} = μ_{Σ}^{(n)} for all n. The second assertion of the proposition will follow immediately by direct computation (remembering μ̃^{(ω)} |_{M} = ν|_{M}). Then the first assertion follows since the expression given for μ_{Σ}^{(n)} is symmetric. We show that λ^{n} defines a f.s. extension in P of μ_{Σ}, and is therefore, by 2.4 and induction equal to μ_{Σ}^{(n)}.

For simplicity of notations, we write the details only for n = 2 (the case n = 1 being true by definition). Let θ(x, y), ϕ(x) ∈ L(M̄). The transformations are explained below.

λ^{2}(θ(x, y) ∧ ϕ(x)) = ∫_{t∈[0,1]} ∫_{t′∈[0,1]} νx̄(θ(x_{t}, x_{t′}) ∧ ϕ(x_{t}))dt′dt

= ∫_{t∈[0,1]} μ_{Σ,y} ⊗ νx̄(θ(x_{t}, y) ∧ ϕ(x_{t}))dt

≤ sup_{a∈ϕ(P)} μ_{Σ,y} (θ(a, y)) × ∫_{t∈[0,1]} νx̄(ϕ(x_{t}))dt

≤ μ_{Σ,x} (ϕ(x)). sup_{a∈ϕ(P)} μ_{Σ,y} (θ(a, y)).

Explanation: ν is a smooth measure, so by 1.10, it admits a unique separated amalgam with any other measure. In particular with μ_{Σ}. The measure ω_{x̄y} defined by ω(θ(x̄, y)) = ∫_{t′∈[0,1]} ν(θ(x̄, x_{t′}))dt is such an amalgam. Therefore it is equal to μ_{Σ,y} ⊗ νx̄. This justifies the second line.

As μ_{Σ} is finitely satisfiable in P, μ_{Σ,y} ⊗ νx̄ is an f.s. extension of μ_{Σ} over ν (in P); this explains the third line. The forth line is just the definition of μ_{Σ}.

If μ is a measure over M, a symmetrization of μ is a global extension μ_{Σ} of μ obtaining by applying the construction above with some μ̃ and ν. If μ is a global M-invariant measure, we will say that μ_{Σ} is a symmetrization of μ over M if it is obtained by the construction above where μ and μ̃ there stand for μ|_{M} and μ respectively.

Note that if we build μ_{Σ} from a Morley sequence of μ̃ over M, then μ_{Σ} is in general not M-invariant. In fact, it is M-invariant if and only if μ is generically stable, in which case μ_{Σ} = μ (as explained below).

Proposition 3.3. Let μ_{x} be a global M-invariant measure, and μ_{Σ} a symmetrization of μ over M. Let also f be an M-definable function whose domain and range is in the same sort as the variable x.

1. If μ is generically stable, then μ_{Σ} = μ, 2. f_{∗}(μ_{Σ}) is a symmetrization of f_{∗}(μ),

3. If f_{∗}(μ) = μ, then f_{∗}(μ_{Σ}) = μ_{Σ}.

Proof. 1. If μ is generically stable, then μ^{(ω)} is totally indiscernible. It follows by 1.6 that for every ϕ(x, c) ∈ L(M̄) and ϵ > 0, the set {t ∈ [0, 1] : |ν(ϕ(a_{t}, c)) − μ̃(ϕ(a, c))| > ϵ} is finite. Therefore the definition of μ_{Σ} implies that μ_{Σ} = μ.

2. Clear: f_{∗}(μ_{Σ}) is a symmetrization of f_{∗}(μ) built using f_{∗}(μ̃) and f_{∗}(ν).

3. Let νx̄ be as in the construction of μ_{Σ}. For I ⊆ [0, 1], define the mea-

sure νI ; x̄ by ν^{I}(ϕ(x_{t_{1}} , .., x_{t_{n}} )) = ν^{I}(ϕ(f_{t_{1}}^{I} (x_{t_{1}} ), .., f_{t_{n}}^{I} (x_{t_{n}} ))) where ϕ(x_{1}, .., x_{n}) ∈

Lx̄(M̄) and f_{t}^{I} = f if t ∈ I and is the identity otherwise.

\underline{Claim}: For every I, ν^{I} is the measure of an M-indiscernible segment.

Proof. The claim concerns only the restriction to M (indeed to ∅) of the measure ν^{I}. Note that ν|_{M} is just μ^{([0,1])}|_{M}, and as f is M definable, the property we need to check does not depend on the choice of the smooth extension ν. It is enough to show that μ^{(ω)} (θ(x_{1}, .., x_{n})) = μ^{(ω)} (θ(g_{1}(x_{1}), .., g_{n}(x_{n}))), for every θ(x_{1}, ..., x_{n}) ∈ L(M), where g_{i} is either f or the identity. This is easily checked by direct computation and induction on n. For example, to see that μ^{(2)}(θ(f(x_{1}), f(x_{2}))) = μ^{(2)}(θ(x_{1}, x_{2})), write

μ_{x_{1}x_{2}}^{(2)} (θ(f(x_{1}), f(x_{2})) = ∫ μ_{x_{2}}(θ(f(a), f(x_{2}))))dμ_{a}
= ∫ f_{∗}μ_{x_{2}}(θ(b, x_{2}))df_{∗}μ_{b}
= ∫ μ_{x_{2}}(θ(b, x_{2}))dμ_{b}
= μ_{x_{1}x_{2}}^{(2)} (θ(x_{1}, x_{2})).

It follows by Proposition 1.6 that for every ϕ(x) ∈ L(M̄) and ϵ, there can be only finitely many values of t ∈ [0, 1] for which |ν(ϕ(x_{t}))−ν(ϕ(f(x_{t})))| > ϵ. Hence the average of νx̄ is the same as the one of νI ; x̄ for each I. As f_{∗}(μ_{Σ}) is the average of ν^{I} for I = [0, 1], we have f_{∗}(μ_{Σ}) = μ_{Σ}.

As an application, we give a short proof of a result from [6], section 7. (It is not stated explicitly there, as the notion of generically stable measure had not been introduced yet, but is the content of the pages from Lemma 7.1 to Lemma 7.6.) If G is a definable group and M a model, by a measure μ_{x} being G(M)-invariant, we mean that μ concentrates on G (i.e. μ(x ∈ G) = 1) and for each g ∈ G(M) and ϕ(x) a formula, μ(ϕ(x)) = μ(ϕ(g.x)).

Proposition 3.4. Let M be a model, and G a definable group. Assume there is μ ∈ ℳ(M) a G(M)-invariant measure. Then μ extends to a global generically stable G(M)-invariant measure.

Proof. First, find a global extension μ̃ of μ that is G(M)-invariant, and M_{1}- invariant for some small M_{1} ⊃ M. This can be done with Keisler’s smooth measure construction, see [6], 7.6. Let μ_{Σ} be a symmetrization of it over M_{1}. Then by the previous propositions, μ_{Σ} is generically stable and G(M)- invariant.

4 Smooth measures

The goal of this section is to prove that generically stable measures in o-minimal theories or in the theory of the p-adics are smooth.

We start by giving a characterization of smoothness which will be useful for proving that measures are smooth. Let M ⊧ T and let ϕ(x, y) ∈ L(M). For b ∈ M̄, we define the border of ϕ(x, b) (over M) as ∂_{x}^{M}ϕ(x, b) = {p ∈ S_{x}(M) : there are a, a′ ⊧ p such that ϕ(a, b)∧¬ϕ(a′, b) holds }. This is a closed subset of the space of types S_{x}(M). We will often omit x and M in the notation. Note that ∂^{M}ϕ(x, b) depends only on q = tp(b/M), so we will also sometimes write ∂^{M}ϕ(x, q) for ∂^{M}ϕ(x, b).

We will also be needing the notion of a localization of a measure μ_{x} ∈ ℳ(M). Let F ⊆ S_{x}(M) be a closed subspace such that μ(F) > 0. Then we define the localization of μ at F by μ_{F}(ϕ(x)) = μ(ϕ(x) ∩ F))/μ(F) for all ϕ(x) ∈ L(M). This is again a Keisler measure over M.

Lemma 4.1. Let μ ∈ ℳ_{x}(N) and F ⊂ S_{x}(N) a closed subset such that μ(F) > 0. If μ is smooth, then μ_{F} is smooth.

Proof. Assume that μ is smooth. Let ν be a global extension of μ_{F} and let μ̃ be the unique global extension of μ. Then we define a global measure ν′ by ν′(ϕ(x)) = μ̃(ϕ(x) ∩ F^{c}) + ν(ϕ(x) ∩ F).μ(F). This defines an global extension of μ. By smoothness, ν′ = μ̃, and it follows that ν = μ̃_{F}.

Lemma 4.2 (Characterization of smoothness). Let μ ∈ ℳ_{x}(M). Then μ is smooth if and only if μ(∂^{M}ϕ(x, b)) = 0 for all ϕ(x, y) ∈ L(M) and all b ∈ M̄.

Proof. Let ϕ(x, y) and b be as in the statement of the lemma. Let O ⊆ S_{x}(M) be the set of types p such that p ⊢ ϕ(x, b). And let F = ∂^{M}ϕ(x, b). Then for any extension ν of μ, we have μ(O) ≤ ν(ϕ(x, b)) ≤ μ(O) + μ(F). Therefore if μ(F) = 0 for all such ϕ and b, then μ is smooth.

Conversely, assume that μ is smooth and let ϕ(x, y) and b be as above. Let ϵ > 0 and take ν_{i}^{1}(x), ν_{i}^{2}(x), ψ_{i}(y), i = 1, .., n be as in Proposition 1.9. Let i be such that ψ_{i}(b) holds. Then ∂ϕ(x, b) ⊆ ν_{i}^{2}(x) \ ν_{i}^{1}(x). Therefore μ(∂ϕ(x, b)) < ϵ. As this is true for all ϵ > 0, μ(∂ϕ(x, b)) = 0.

To illustrate this, assume T is o-minimal, let M ≺ N be models of T and let ϕ(x) ∈ L(N) be a formula, x a single variable. By o-minimality, ϕ(x) is a finite union of (closed or open) intervals. Let a_{0}, .., a_{n−1} denote those end points that lie in N \ M. Then ∂^{M}ϕ = {tp(a_{k}/M) : k < n}. In particular, it is finite.

Lemma 4.3. Let μ_{x} be a global measure, smooth over M. Let f be an M definable function whose range is the sort of the variable x. Then f_{∗}(μ) is smooth.

Proof. Let μ_{x} be a global measure, smooth over M. Let λ_{y} = f_{∗}(μ) and let ϕ(y) be an M-definable set. Let F = ∂^{M}ϕ(y). Define also ψ(x) = ϕ(f(x)) and G = ∂^{M}ψ(x). Then λ(F) = μ(G) = 0 as μ is smooth. By Lemma 4.2, f_{∗}(μ) is smooth.

The following easy fact will be used implicitly in what follows.

Lemma 4.4. Let μ be a global smooth measure. Assume that μ is M-invariant, then μ is smooth over M.

Proof. Proposition 1.9 gives us formulas ψ_{i}(y, d), ν_{i}^{1}(x, d) and ν_{i}^{2}(x, d) with d ∈ M̄ satisfying three properties as stated. It is enough to show that we can find d′ ∈ M such that the formulas ψ_{i}(y, d′), ν_{i}^{1}(x, d′), ν_{i}^{2}(x, d′) satisfy the same properties. Now the condition imposed on d′ by the first 2 properties is clopen. As μ is definable over M, the condition imposed by the third point is open. So we are looking for d′ in some open set of S(M). As we know that this set is non-empty, it must intersect the set of realized types, and we find the required d′.

Lemma 4.5. Let S_{1}, S_{2} be two sorts and f : S_{1} ) S_{2} a surjective definable map. Assume that all generically stable measures on S_{1} are smooth, then it is also the case for S_{2}.

Proof. Let μ_{x} a generically stable measure concentrating on S_{2} (i.e., x is a variable of sort S_{2}). Let M be a small model such that μ is M-invariant. We can find a measure η_{y} on M̄ which is M-invariant and such that f_{∗}(η) = μ (this can be proven using 1.1. Take Ω_{1} to be definable sets of the form f^{−1}(ψ), ψ a definable set of S_{2}. Let Ω_{2} be composed of formulas ϕ(x) ∈ L(M̄) that fork over M. Define μ_{0} on Ω = ⟨Ω_{1}, Ω_{2}⟩ by μ_{0} = f^{−1}(μ) on Ω_{1} and μ_{0} = 0 on Ω_{2}. By the Lemma extend μ_{0} to a global measure). Consider a symmetrization η_{Σ} of η over M. Then f_{∗}(η_{Σ}) is a symmetrization of μ over M, and therefore is equal to μ.

By hypothesis, η_{Σ} is smooth. So μ = f_{∗}(η_{Σ}) is also smooth.

We will use this ad-hoc criterion for smoothness.

Proposition 4.6. Assume T has definable Skolem functions. Let 𝒮 be a set of imaginary sorts containing the main sort M̄. Assume that for any model N, any formula ϕ(x̄, y) ∈ L(N) (y a single variable from the main sort) is a boolean combination of formulas of the form R(f(x̄), y) where R is a ∅-definable relation and f is an N-definable function taking values in a sort from 𝒮. Assume that for each S ∈ 𝒮, all generically stable measures over S are smooth.

Then any generically stable measure is smooth.

Proof. Let M ≺ N be two models of T. Assume that for all n, all n types over M are realized in N. Let μ ∈ ℳ(N) be an M-invariant generically stable measure in k variables. It is enough to show that any formula of the form ϕ(x̄, c), c ∈ M̄ a 1-tuple and ϕ(x̄, y) ∈ L(N), has the same measure in any extension of μ. (Because then N(c) is a model of T over which μ has a unique extension, and we can replace N by it and iterate.) By hypothesis, we may assume that ϕ(x̄, y) = R(f(x̄), y) for some R and f as above.

If ν is a global extension of μ, then ν(ϕ(x̄, c)) = f_{∗}(ν)(R(z, c)). Now f_{∗}(ν) is generically stable and N-invariant. By hypothesis, it is smooth over N. Therefore ν(ϕ(x̄, c)) is determined.

Corollary 4.7. If T is o-minimal, then any generically stable measure is smooth.

Proof. We will check the hypothesis of the previous proposition for 𝒮 = {M̄} (the main sort). Let T be o-minimal. Every formula ϕ(x̄, y) is a boolean combination of formulas of the form y < f(x̄) and y = f(x̄) where f is a definable function. So the first part of the hypothesis is satisfied. Next, consider μ_{x} a global generically stable measure in dimension 1. Let ϕ(x) be a formula, x a single variable, with parameters in some extension N̄ ≻ M̄. As explained after the proof of 4.2, ∂M̄ϕ is a finite set of non-realized types. Finiteness easily implies that all the types in ∂M̄ϕ are generically stable. As there are no non-realized generically stable types in T, μ(∂M̄ϕ) = 0. By 4.2, μ is smooth. Proposition 4.6 therefore applies.

Corollary 4.8. Let T = Th(ℚ_{p}), for some p, then any generically stable measure is smooth.

Proof. This case is similar to the o-minimal one. Let Γ denote the value group. For 1 ≥ k, n < ω, let 𝔅_{k,n} be the set of canonical parameters of definable sets of the form {x : val(x − a) ≡ k [n]}, a ∈ M̄ and 𝔅 be the set of canonical parameters of balls {x : val(x − a) = α} for a ∈ M̄, α ∈ Γ.

We will check that Proposition 4.6 applies with 𝒮 = ⋃_{k,n} 𝔅_{k,n} ∪ {M̄, 𝔅}. We leave it to the reader to check that all generically stable measures in one

variable from M̄ or from Γ are smooth (this can be done as in the o-minimal case: check that the border of a definable set is finite).

Next, let μ_{x} be a generically stable measure on 𝔅. Let val denote the natural map from 𝔅 to Γ. Then, val_{∗}(μ) is generically stable and therefore is smooth. We may assume that val_{∗}(μ) is either a realized type or an atomless measure. Assume val_{∗}(μ) = ‘‘x = α" for some α ∈ Γ. Then μ is a measure concentrating on 𝔅_{α}: the sort of balls of radius α. There is a surjective map π : M̄ ) 𝔅_{α}, so by Lemma 4.5, μ is smooth. Now if val_{∗}(μ) is atomless, one can check by inspection that μ(∂ϕ(x)) = 0 for every definable set ϕ(x).

Finally, for any k, n < ω, there is a surjective map from 𝔅 to 𝔅_{k,n}, so all generically stable measures there are also smooth.

Let ϕ(x) ∈ L(A) be a definable set in dimension 1. Then ϕ(x) can be written as a finite boolean combination of formulas of the form x ∈ b with b in some 𝔅_{k,n}, k, n < ω or in 𝔅. We can choose the decomposition such that b is A-definable. Therefore Proposition 4.6 applies.

Theories in which all generically stable measures are smooth will be studied in a subsequent work [13], where equivalent characterizations will be given along with some properties. In particular, it will be shown that in a dpminimal theory with no generically stable type in the main sort, all generically stable measures are smooth, generalizing the two corollaries above.

5 σ-additive measures

We prove in this section the main result of this paper: σ-additive measures are generically stable.

Recall that if M ⊧ T, an externally definable subset of M is a subset of the form ϕ(M) where ϕ(x) ∈ L(M̄). Assume that the model M is equipped with a σ-algebra 𝒜 such that externally definable sets are measurable. Then if μ_{0} is a (σ-additive) probability measure on (M, 𝒜), μ_{0} induces a global Keisler measure μ. Namely μ(ϕ(x)) = μ_{0}(ϕ(M)), for ϕ(x) ∈ L(M̄).

Theorem 5.1. Let T be NIP, M ⊧ T equipped with a σ-algebra 𝒜. Assume that any externally definable subset of M^{2} is measurable for the product algebra 𝒜^{⊗2}. Let μ_{0} be a probability measure on (M, 𝒜). Then μ_{0} induces a global Keisler measure μ and μ is generically stable.

Proof. Note first that by construction μ is finitely satisfiable in M (hence also M-invariant).

We have at our disposal two different amalgams of μ_{x} by μ_{y}. The first one is μ_{xy}^{(2)} = μ_{x} ⊗ μ_{y} from the model-theoretic world. The second one comes from probability theory: we may form the product measure μ_{0}^{2} = μ_{0} × μ_{0} which is a σ-additive measure on (M^{2}, 𝒜^{⊗2}). By hypothesis, μ_{0}^{2} induces a global Keisler measure μ_{2}. Of course, μ^{(2)} and μ_{2} coincide on products ϕ(x)∧ψ(y) (they are both separated amalgam of μ_{x} and μ_{y}). We will prove that in fact μ_{2} = μ^{(2)}. For this, it is enough to check that μ_{2} is an f.s. extension in M of μ_{x}.

Let θ(x, y), ϕ(x) ∈ ℒ(M̄). We have:

μ_{0}^{2}(θ(x, y) ∧ ϕ(x)) = ∫_{(a,b)∈M^{2}} θ(a, b) ∧ ϕ(a)dμ_{0}^{2}

= ∫_{a∈ϕ(M)} ∫_{b∈M} θ(a, b)dμ_{0}dμ_{0}

≤ μ_{0}(ϕ(M)). sup_{a∈ϕ(M)} ∫_{b∈M} θ(a, b)dμ_{0}

= μ(ϕ(x)). sup_{a∈ϕ(M)} μ(θ(a, y)).

By Proposition 2.4, this proves that μ_{2} = μ_{x} ⊗ μ_{y}.

By the usual Fubini theorem, μ_{0}^{⊗2} is a symmetric measure. Therefore it is also the case for μ^{(2)}: μ^{(2)}(θ(x, y)) = μ^{(2)}(θ(y, x)) for all θ(x, y) ∈ L(M̄). By Theorem 1.7, this shows that μ is generically stable.

Remark 5.2. The assumption that externally definable sets are measurable for the product σ-algebra is of course necessary. Consider for example an ω_{1}- saturated model M of RCF. Let p ∈ S(M) be the type at + and p̃ the global co-heir of p. Then p̃ is induced by a σ-additive measure on M (equipped with the Borel σ-algebra). Of course it is not generically stable, and note that the set {(x, y) ∈ M^{2} : x ≤ y} is not measurable for the product algebra.

As a corollary, we recover the following result from [7], Section 6.

Corollary 5.3. Let M be either ℝ: the standard real numbers equipped with any o-minimal structure expanding the field operations, or ℚ_{p}: the standard p-adic field. Let μ_{0} be a σ-additive measure on M, then μ_{0} induces a smooth Keisler measure μ.

Proof. Theorem 5.1 implies that μ is generically stable, then using Corollary 4.7 or 4.8, we deduce that it is smooth.

Question 5.4. More generally, if we assume that 𝒜 is generated as a σ-algebra by definable sets, is it the case that μ is smooth?

Proposition 5.5. Let M = ℝ be an o-minimal expansion of the standard model. Let μ be a smooth measure over M^{k} concentrating on [0, 1]^{k}. Then there is a σ-additive Borel measure μ_{0} on [0, 1]^{k} such that μ_{0} induces μ as a Keisler measure.

Proof. Let Ω_{k} be the subspace of S_{k}(M) of types that concentrate on [0, 1]^{k}. Note that the standard part application st induces an application st : Ω_{k} ) [0, 1]^{k}. This is a Borel map (the inverse image of a closed set is closed). In particular, we can consider the pushforward measure μ_{0} = st_{∗}(μ). It is a σadditive measure on [0, 1]^{k}. All we now need to show is that μ_{0} induces μ as a Keisler measure, i.e., that μ(X) = μ_{0}(X) for any definable set X.

Let X be a definable set of [0, 1]^{k}. Assume first that X is closed. We have a definable map d_{X} : M^{k} ) M such that d_{X}(x̄) is the distance of x̄ to X. For ϵ > 0, let X_{ϵ} be the closed ϵ-neighborhood of X: X_{ϵ} = {x̄ ∈ [0, 1]^{k} : d_{X}(x̄) ≤ ϵ}. It is also a definable set. We have st^{−1}(X) = ∩_{ϵ>0} X_{ϵ}. Let p be a type in st^{−1}(X)\X. Then (d_{X})_{∗}(p) is the type 0^{+} of S(M). As μ is smooth, (d_{X})_{∗}(μ) is also smooth and it is not possible that it has an atom on 0^{+}. Thus μ(st^{−1}(X) \ X) = 0.

We treat the general case by induction on the dimension of X. The case of dimension 0 is trivial. So let X be any definable set. Let O be its interior and X̄ its closure. Then D = X̄\ O has lower dimension then X. We know that μ(X̄) = μ_{0}(X̄) and by induction μ(X̄\X) = μ_{0}(X̄\X). Hence μ(X) = μ_{0}(X).

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