2. Keisler measures and ergodic measures

We begin with a brief and self-contained introduction to ergodic measures and Keisler measures. Firstly, we introduce ergodic measures in a general context. Then, we explain their utility for the study of Keisler measures.

2.1. Ergodic measures

Ergodic measures are an essential tool in our paper. They are better behaved than invariant measures and any invariant measure can be decomposed as an integral average of ergodic measures. Here we briefly introduce these measures and mention some basic results about them. Chapter 12 of Phelps’ book [Reference Phelps19] covers the theory we discuss at the adequate level of generality for our purposes.

We work in the context of a topological group G acting on a topological space X via a continuous action $\cdot :G\times X\to X$ . When X is compact and Hausdorff, we call this action a ${G}$ -flow. Let $\mathcal {B}(X)$ be the set of Borel subsets of X, and let $\mu :\mathcal {B}(X)\to [0,1]$ be a Borel probability measure. We say that $\mu $ is ${G}$ -invariant if for any $\tau \in G$ and any $A\in \mathcal {B}(X)$ , we have that $\tau \cdot A\in \mathcal {B}(X)$ and $\mu (\tau \cdot A)=\mu (A)$ .

Definition 2.1. We say that a G-invariant Borel probability measure $\mu $ is ${G}$ -ergodic if for all $A\in \mathcal {B}(X)$ , we have that if for all $\tau \in G$ ,

$$\begin{align*}\mu(A \bigtriangleup \tau \cdot A)=0, \end{align*}$$

then either $\mu (A)=0$ or $\mu (A)=1$ .

An alternative definition tells us that any invariant function is constant [Reference Einsiedler6, Proposition 2.14]:

Proposition 2.2. Let $(X, \mathcal {B}, \mu )$ be a probability space and G be a group acting on X such that $\mu $ is G-invariant. Then, the following are equivalent $:$

1. 1. The measure $\mu $ is G-ergodic.

2. 2. Any measurable function $f:X\to \mathbb {C}$ , which is G-invariant almost everywhere $($ i.e., for any $\tau \in G, f\circ \tau =f$ a.e. $)$ is constant almost everywhere.

We are interested in studying G-ergodic measures since, when X is metrizable, they yield an ergodic decomposition of any G-invariant measure. Hence, their study is essential to the understanding of the G-invariant measures on X. Below, for $f\in C(X, \mathbb {C})$ , we write $\mu (f)=\int _X f \mathrm {d}\mu $ . From [Reference Phelps19, p. 77] we have:

Theorem 2.3 (Ergodic decomposition)

Let X be a compact metrizable space with a group G acting continuously on it. Let $\mu $ be a G-invariant Borel probability measure on X. Then, the space $\mathfrak {M}(X)$ of G-invariant Borel probability measures on X is also metrizable and the set of G-ergodic measures $\mathrm {Erg}(X)$ is a Borel subset of $\mathfrak {M}(X)$ . Furthermore, there is a unique Borel probability measure $\mathfrak {m}$ on $\mathfrak {M}(X)$ such that for any $f\in C(X, \mathbb {C})$ ,

$$\begin{align*}\mu(f)=\int_{\mathrm{Erg}(X)} \nu(f) \mathrm{d}\mathfrak{m}(\nu).\end{align*}$$

2.2. Keisler measures

Keisler measures are finitely additive probability measures on the space of definable subsets of a structure. Chapter 7 of Simon’s book [Reference Simon21] is a good introduction to the subject. We give a brief self-contained discussion and note the importance of ergodic measures in the study of invariant Keisler measures.

Let T be a complete $\mathcal {L}$ -theory, where $\mathcal {L}$ is a countable first-order language and $\mathcal {M}\models T$ . Let $\mathrm {Def}_x(M)$ denote the Boolean algebra of $\mathcal {L}(M)$ -formulas in the free variable x up to $\mathrm {Th}(\mathcal {M}_M)$ -equivalence, where $\mathcal {M}_M$ is the expansion of $\mathcal {M}$ by constant symbols for each element of M. Let $S_x(M)$ be the Stone space of types over M in the variable x with the usual topology. We write $[\phi (x,a)]$ for the clopen set of types containing the formula $\phi (x,a)$ .

This space is always compact and Hausdorff. Moreover, when M is countable, it is also metrizable. This can be seen by taking an enumeration of the $\mathcal {L}(M)$ -formulas $\phi _1, \phi _2, \dots $ and then for $p_1, p_2\in S_x(M),$ letting $d(p_1,p_2)=\frac {1}{2^n}$ , where n is the smallest natural number such that $\phi _n$ is not contained in both $p_1$ and $p_2$ . The automorphism group $\mathrm {Aut}(M)$ naturally acts on $\mathrm {Def}_x(M)$ and on $S_x(M)$ , where for $\tau \in \mathrm {Aut}(M)$ , $\tau \cdot \phi (x, a)=\phi (x, \tau (a))$ and for $p\in S_x(M)$ ,

$$\begin{align*}\tau\cdot p=\{\phi(x, \tau(a)) : \phi(x, a)\in p\}.\end{align*}$$

Definition 2.4. Let $\mathcal {M}$ be an $\mathcal {L}$ -structure. We say that $\mu :\mathrm {Def}_x(M)\to [0,1]$ is a Keisler measure if it is a finitely additive measure such that $\mu (x=x)=1$ . We say that $\mu $ is $\mathrm {Aut}(M)$ -invariant if for any $\tau \in \mathrm {Aut}(M)$ ,

$$\begin{align*}\mu(\tau\cdot \phi(x, a))=\mu(\phi(x, a)).\end{align*}$$

For an infinite cardinal $\kappa $ , we say that $\mathcal {M}$ is strongly $\boldsymbol {\kappa }$ -homogeneous if whenever a and $a'$ are tuples of cardinality $<\kappa $ such that $a\equiv a'$ , there is $\tau \in \mathrm {Aut}(M)$ such that $\tau (a)=a'$ . If $\mathcal {M}$ is strongly $\omega $ -homogeneous, for a Keisler measure to be $\mathrm {Aut}(M)$ -invariant is equivalent to saying that, if $a\equiv a'$ , then $\mu (\phi (x, a))=\mu (\phi (x, a'))$ .

As noted in the introduction, Keisler measures give us a natural notion of smallness for a definable set.

Definition 2.5. Let $\mathcal {M}$ be $\omega $ -saturated and strongly $\omega $ -homogeneous. We say that $\phi (x, b)\in \mathrm {Def}_x(M)$ is universally measure zero if $\mu (\phi (x,b))=0$ for every $\mathrm {Aut}(M)$ -invariant Keisler measure. Similarly, let $A\subseteq M$ and let $\mathcal {M}$ be $\kappa $ -saturated and strongly $\kappa $ -homogeneous for $\kappa =\max \{|A|^+, \omega \}$ . We say that $\phi (x, b)\in \mathrm {Def}_x(M)$ is universally measure zero over ${A}$ if it is assigned measure zero by every $\mathrm {Aut}(M/A)$ -invariant Keisler measure. We call $\mathcal {O}(A)$ the set of formulas in $\mathrm {Def}_x(M)$ which are universally measure zero over A.

The set $\mathcal {O}_x(A)=\mathcal {O}(A)\cap \mathrm {Def}_x(M)$ forms an ideal in the Boolean algebra of $\mathrm {Def}_x(M)$ [Reference Chernikov, Hrushovski, Kruckman, Krupiński, Moconja, Pillay and Ramsey3]. Similarly, $F_x(A)$ , the set of formulas in the variable x with parameters from M forking over A, also forms an ideal. Our main result is that there are $\omega $ -categorical supersimple theories for which $\mathcal {O}(\emptyset )$ strictly contains the set of formulas forking over $\emptyset $ , $F(\emptyset )$ .

Remark 2.6. In other articles [Reference Chernikov, Hrushovski, Kruckman, Krupiński, Moconja, Pillay and Ramsey3], being universally measure zero is defined in the monster model for the theory. In this article, since we work over finite sets, no generality is lost by focusing on an $\omega $ -saturated strongly $\omega $ -homogeneous model. In fact, if $F(\emptyset )=\mathcal {O}(\emptyset )$ in an $\omega $ -saturated strongly $\omega $ -homogeneous model of T, this is the case in all such models (cf. [Reference Hrushovski, Krupiński and Pillay12, Proposition 2.3]). In particular, we may focus on invariant Keisler measures on the countable model of the $\omega $ -categorical theory we are studying. More generally, whenever we compare the sizes of $F(A)$ and $\mathcal {O}(A)$ in this article we always work over a sufficiently saturated and strongly homogeneous model.

A Keisler measure $\mu $ can be extended uniquely to a regular $\sigma $ -additive Borel probability measure $\mu : \mathcal {B}_x(M)\to [0,1]$ , where $\mathcal {B}_x(M)$ is the set of Borel subsets of $S_x(M)$ [Reference Simon21, Section 7.1]. Conversely, any regular Borel probability measure on $S_x(M)$ induces a Keisler measure by considering its restriction to clopen sets. This correspondence still holds between $\mathrm {Aut}(M)$ -invariant Keisler measures in the variable x and $\mathrm {Aut}(M)$ -invariant regular Borel probability measures on $S_x(M)$ . From here, we shall speak interchangeably of the two.

Remark 2.7. An $\mathrm {Aut}(M^{eq})$ -invariant Keisler measure $\mu ^{eq}$ on $\mathcal {M}^{eq}$ in the real variable x is entirely determined by its restriction to its induced $\mathrm {Aut}(M)$ -invariant Keisler measure $\mu $ on $\mathcal {M}$ . Furthermore, $\mu ^{eq}$ is ergodic if and only if $\mu $ is.

If we are interested in studying the universally measure zero formulas for a countable structure, it is helpful to study $\mathrm {Aut}(M)$ -ergodic measures on $S_x(M)$ . From the ergodic decomposition 2.3, we get:

Corollary 2.8. Let $\mathcal {M}$ be a countable structure and let $\mu $ be an $\mathrm {Aut}(M)$ -invariant Borel probability measure on $S_x(M)$ . Let $\mathfrak {M}_x(M)$ be the space of $\mathrm {Aut}(M)$ -invariant Borel probability measures on $S_x(M)$ . Then, there is a unique Borel probability measure $\mathfrak {m}$ on $\mathfrak {M}_x(M)$ such that for any $\mathcal {L}(M)$ -formula $\phi (x,a)$ ,

$$\begin{align*}\mu([\phi(x,a)])=\int_{\mathrm{Erg}_x(M)} \nu([\phi(x,a)]) \mathrm{d}\mathfrak{m}(\nu),\end{align*}$$

where $\mathrm {Erg}_x(M)$ is the space of $\mathrm {Aut}(M)$ -ergodic measures.

Remark 2.9. When M is countable, every Borel measure $\mu $ on $S_x(M)$ is regular [Reference Bogachev1, Theorem 7.1.7]. From [Reference Evans and Tsankov9, Corollary 1.2], we actually know that when $\mathcal {M}$ is the countable model of an $\omega $ -categorical theory, $\mathrm {Erg}_x(M)$ is closed in the space $\mathfrak {M}_x(M)$ .

For conciseness of notation, in subsequent sections we shall generally refer to $\mu $ as an ergodic measure on $\mathcal {M}^{eq}$ (in the variable x). By this we mean that $\mu $ is a Borel probability measure on $S_x(M^{eq})$ invariant under the action of $\mathrm {Aut}(M^{eq})$ , which is also $\mathrm {Aut}(M^{eq})$ -ergodic. As noted above, when x is a variable in the real sort, there is a one-to-one correspondence between these ergodic measures and the ergodic $\mathrm {Aut}(M)$ -invariant Borel probability measures on $S_x(M)$ . More generally, given the various correspondences explained in this section we can safely use $\mu $ to denote both the Keisler measure and the corresponding Borel probability measures on $S_x(M)$ and $S_x(M^{eq})$ .

3. Weak algebraic independence and probabilistic independence

Given the Borel probability space $(S_x(M^{eq}), \mathcal {B}_x(M^{eq}), \mu )$ induced by an invariant Keisler measure on M in the variable x, we wish to study the associated Hilbert space $L^2(\mu )$ . In this section we introduce some of the relevant tools for this, following the discussion of [Reference Jahel and Tsankov13, Reference Tsankov23]. Furthermore, we shall show how various results of [Reference Jahel and Tsankov13] yield that a weak form of algebraic independence between parameters implies a form of probabilistic independence in an ergodic measure. This result is Corollary 3.8.

Let $\mathcal {M}$ be a countable $\omega $ -categorical structure. We have that $\mathrm {Aut}(M)$ is a Polish group, that is, a separable completely metrizable topological group. In particular, in the topology of $G=\mathrm {Aut}(M)$ , the pointwise stabilizers of finite sets $G_A=\mathrm {Aut}(M/A)$ for $A\subset M$ finite, are neighbourhoods of the identity and the set of cosets of these pointwise stabilizers form a basis of open sets for the topology [Reference Macpherson15, Section 4.1].

As noted in the previous section, a Keisler measure $\mu $ on $\mathcal {M}$ in the variable x induces a Borel probability space $(S_x(M^{eq}), \mathcal {B}_x(M^{eq}), \mu )$ . We are interested in studying the complex $L^2$ -space $L^2(S_x(M^{eq}), \mathcal {B}_x(M^{eq}), \mu )$ , which we abbreviate $L^2(\mu )$ .

The action of $\mathrm {Aut}(M)$ on M naturally induces an action $\lambda : \mathrm {Aut}(M)\times L^2(\mu )\to L^2(\mu )$ , where for $\sigma \in \mathrm {Aut}(M), f\in L^2(\mu )$ ,

$$\begin{align*}\lambda(\sigma, f)(p)=f(\sigma^{-1}(p)) \text { for all } p\in S_x(M).\end{align*}$$

This action is well defined and preserves integrals. For each $\sigma \in \mathrm {Aut}(M)$ , the map $\Lambda _\sigma :L^2(\mu )\to L^2(\mu )$ given by $f\mapsto \lambda (\sigma , f)$ is a unitary operator. Moreover, the map $\pi : \mathrm {Aut}(M) \to U(L^2(\mu ))$ from the group of automorphisms of M to the group of unitary operators of $L^2(\mu )$ given by $\sigma \mapsto \Lambda _\sigma $ is a unitary representation (cf. Section 3 of [Reference Tsankov23]).

Let $\mathcal {H}$ be a complex Hilbert space and the action of $G=\mathrm {Aut}(M)$ on $\mathcal {H}$ be a unitary representation. For $A\subseteq M^{eq}$ we write, following [Reference Jahel and Tsankov13],

$$\begin{align*}\mathcal{H}_A=\overline{\{f\in \mathcal{H} | G_{A'}\cdot f=f \text{ for some finite } A'\subseteq A\}},\end{align*}$$

where for $S\subset \mathcal {H}$ , $\overline {S}$ is its closure.

Note that when the $\mathcal {H}_i$ are all closed subspaces for $i\in \{1, 2, 3\}$ with $\mathcal {H}_1\perp _{\mathcal {H}_0}\mathcal {H}_2$ , we have that for $f\in \mathcal {H}_1$ we can decompose f into $f=f_1+f_0$ , where $f'\in \mathcal {H}_1\ominus \mathcal {H}_0$ and $f_0\in \mathcal {H}_0$ . Similarly, for $g\in \mathcal {H}_2$ we can decompose it as $g=g_2+g_0$ . Now, by the orthogonalities $(\mathcal {H}_i\ominus \mathcal {H}_0)\perp \mathcal {H}_0$ for $i\in \{1,2\}$ and $(\mathcal {H}_1\ominus \mathcal {H}_0)\perp (\mathcal {H}_2\ominus \mathcal {H}_0)$ , we have that

$$\begin{align*}\langle f, g \rangle=\langle f_0, g_0 \rangle,\end{align*}$$

where $\langle \cdot , \cdot \rangle $ is the inner product on $\mathcal {H}$ .

In an $\omega $ -categorical structure, we have the following theorem of Jahel and Tsankov, which translates weak algebraic independence into orthogonality of the associated Hilbert spaces:

Note that the space of constant functions is closed in $L^2(\mu )$ ; hence, from Proposition 2.2 we get:

An ergodic measure also concentrates on an orbit. So, for ergodic $\mu $ , in $L^2(\mu )$ the constant function $\mathbb {1}$ will be in the same equivalence class as $\mathbb {1}_\phi $ , where $\phi $ isolates one of the finitely many types over the empty set, by Ryll-Nardzewski.

Theorem 3.3 yields very powerful results when considering an ergodic measure and the inner product on $L^2(\mu )$ .

Theorem 3.7. Let $\mathcal {M}$ be an $\omega $ -categorical countable structure with $\mathrm {acl}^{eq}(\emptyset )=\mathrm {dcl}^{eq}(\emptyset )$ . Consider the complex Hilbert space $\mathcal {H}=L^2(\mu )$ , where $\mu $ is an ergodic measure. Suppose that $A, B \subseteq M^{eq}$ are algebraically closed and that . Let $f\in \mathcal {H}_A, g\in \mathcal {H}_B$ . Then,

$$\begin{align*}\langle f, g \rangle=\langle f, \mathbb{1}\rangle \overline{\langle g, \mathbb{1}\rangle}.\end{align*}$$

Proof Let $f_0$ and $g_0$ be the projections of f and g, respectively, on $\mathcal {H}_\emptyset $ . Now, since $f\perp _{\mathcal {H}_\emptyset } g$ , $\langle f, g \rangle =\langle f_0, g_0 \rangle $ . Since $f_0, g_0\in \mathcal {H}_\emptyset $ , by Lemma 3.5, $f_0=\alpha \mathbb {1}$ , $g=\beta \mathbb {1}$ for $\alpha , \beta \in \mathbb {C}$ . Now, this yields that

$$\begin{align*}\langle f, g\rangle=\langle f_0, g_0 \rangle=\langle \alpha \mathbb{1},\beta \mathbb{1} \rangle=\alpha \overline{\beta}\langle \mathbb{1}, \mathbb{1}\rangle.\end{align*}$$

Being in a probability space, $\langle \mathbb {1}, \mathbb {1}\rangle =1$ . However, since $\langle f-f_0, f_0\rangle =0$ , we obtain that $\langle f, \mathbb {1} \rangle =\alpha $ , and similarly, $\langle g, \mathbb {1} \rangle =\beta $ . This yields the desired result.

This is already substantially observed in Corollary 3.5 and Remark 3.6 of [Reference Jahel and Tsankov13]. In particular, since we are interested in the measures of formulas, we have that:

Corollary 3.8. Let $\mathcal {M}$ be $\omega $ -categorical. Let $\mu $ be an ergodic measure on ${M}^{eq}$ in the variable x. Suppose that $\mathrm {acl}^{eq}(\emptyset )=\mathrm {dcl}^{eq}(\emptyset )$ . Let $a, {b}$ be tuples from $\mathcal {M}^{eq}$ such that . Then, for any $\mathcal {L}^{eq}$ -formulas $\phi (x, {y}), \psi (x, {z})$ ,

$$\begin{align*}\mu(\phi(x, {a}) \wedge \psi(x, {b}))=\mu(\phi(x, {a}) )\mu(\psi(x, {b})).\end{align*}$$

Proof Note that the indicator functions $\mathbb {1}_{\phi (x, {a})}$ and $\mathbb {1}_{\psi (x, {b})}$ are in $\mathcal {H}_{\mathrm {acl}^{eq}({a})}$ and $\mathcal {H}_{\mathrm {acl}^{eq}({b})}$ , respectively, and that

$$ \begin{align*} \mu(\phi(x, {a})\cap \psi(x, {b})) & = \int_{S_x(M)} \mathbb{1}_{\phi(x, {a})\cap\psi(x, {b})} \mathrm{d}\mu\\ & =\int_{S_x(M)} \mathbb{1}_{\phi(x, {a})}\cdot \mathbb{1}_{\psi(x, {b})} \mathrm{d}\mu \\ & = \left(\int_{S_x(M)} \mathbb{1}_{\phi(x, A)}\mathrm{d}\mu\right)\overline{\left(\int_{S_x(M)} \mathbb{1}_{\phi(x, {a})}\mathrm{d}\mu\right)} \\ &=\mu(\phi(x, {a}))\overline{\mu(\psi(x, {b}))} \\ &= \mu(\phi(x, {a}))\mu(\psi(x, {b})). \end{align*} $$

Here the third equality holds by Theorem 3.7 and the last one follows since our measure is real-valued. Hence, the result follows.

Remark 3.9. This corollary has a partial converse which is well known in ergodic theory [Reference Jahel and Tsankov13, Proposition 4.8]. Fix an invariant Keisler measure $\mu $ and suppose that for any formula $\phi (x,a)$ there is an automorphism $\sigma \in \mathrm {Aut}(M)$ such that

$$\begin{align*}\mu(\phi(x,a)\wedge \phi(x, \sigma \cdot a))=\mu(\phi(x,a))^2.\end{align*}$$

Then, $\mu $ is ergodic.

We conclude this section with a brief discussion of the assumption of $\mathrm {acl}^{eq}(\emptyset )=\mathrm {dcl}^{eq}(\emptyset )$ in Theorem 3.7 and Corollary 3.8. The assumption is needed in order to have the equality $L^2(\mu )_{\mathrm {acl}^{eq}(\emptyset )}= L^2(\mu )_{\mathrm {dcl}^{eq}(\emptyset )}$ . The latter then yields the desired results since we know from Corollary 3.5 that $L^2(\mu )_{\mathrm {dcl}^{eq}(\emptyset )}$ is generated by the constant indicator function. However, we can also obtain the relevant independence results when $\mathrm {acl}^{eq}(\emptyset )\supsetneq \mathrm {dcl}^{eq}(\emptyset )$ . The following lemma is common knowledge:

Lemma 3.10. Let $\mathcal {M}$ be $\omega $ -categorical. Let $A\subseteq M$ be finite. Fix the variable x in the home sort. Then, there is $A_0\subseteq \mathrm {acl}^{eq}(A)$ finite such that for every b, a tuple from $\mathcal {M}$ in the variable x,

(3.1) $$ \begin{align} \mathrm{tp}(b/A_0)\vdash \mathrm{tp}(b/\mathrm{acl}^{eq}(A)). \end{align} $$

By this we mean that if b and $b'$ in the variable x have the same type over $A_0$ , then they have the same type over $\mathrm {acl}^{eq}(A)$ in $\mathcal {M}^{eq}$ .

A useful consequences of the lemma is:

Corollary 3.11. Let $\mathcal {M}$ be $\omega $ -categorical. Let $\mu $ be an invariant Keisler measure on $\mathcal {M}^{eq}$ in the variable x. Then, there is finite $A_0\subseteq \mathrm {acl}^{eq}(\emptyset )$ such that

$$\begin{align*}L^2(\mu)_{A_0}=L^2(\mu)_{\mathrm{acl}^{eq}(\emptyset)}.\end{align*}$$

For an $\omega $ -categorical structure $\mathcal {M}$ and given the variable x and $A_0$ as above, there are finitely many types over $A_0$ in the variable x, isolated by formulas $\chi _1(x), \dots , \chi _m(x)$ . By additivity, any invariant Keisler measure $\mu $ can be written as a weighted sum of measures $\mu _{\chi _i}$ for $1\leq i\leq m'\leq m$ , where, for $\mu (\chi _i(x))>0$ , $\mu _{\chi _i}$ is the $\mathrm {Aut}(M/A_0)$ -invariant Keisler measure induced from $\mu $ by

$$\begin{align*}\mu_{\chi_i}(\phi(x,a))=\frac{\mu(\phi(x,a)\wedge \chi_i(x))}{\mu(\chi_i(x))}.\end{align*}$$

From Corollary 3.11, we know that for any $\mathrm {Aut}(M/A_0)$ -ergodic measure $\nu $ we have that for , and $\mathcal {L}^{eq}$ -formulas $\phi (x,y), \psi (x, z)$ ,

$$\begin{align*}\nu(\phi(x,a)\wedge \psi(x, b))=\nu(\phi(x,a))\nu(\psi(x, b)).\end{align*}$$

Hence, in the context of $\mathrm {acl}^{eq}(\emptyset )\supsetneq \mathrm {dcl}^{eq}(\emptyset )$ , in order to study invariant Keisler measures on an $\omega $ -categorical structure $\mathcal {M}$ we may naturally move to study the $\mathrm {Aut}(M/A_0)$ -invariant Keisler measures.

4. A strong independence theorem

In this section, we prove that in a simple $\omega $ -categorical structure $\mathcal {M}$ with $\mathrm {acl}^{eq}(\emptyset )=\mathrm {dcl}^{eq}(\emptyset )$ if forking over the empty set is the same as being universally measure zero, then $\mathcal {M}$ satisfies a stronger version of the independence theorem over finite algebraically closed sets. We conclude with some consequences for $\omega $ -categorical $MS$ -measurable structures. Unless specified otherwise, $\mathcal {M}$ denotes the countable model of an $\omega $ -categorical theory.

Firstly, we note how the measure $\mu (\phi (x, a)\wedge \psi (x, b))$ only depends on the types of the individual parameters when a and b are weakly algebraically independent.

Corollary 4.1. Let $\mathcal {M}$ be an $\omega $ -categorical structure with $\mathrm {acl}^{eq}(\emptyset )=\mathrm {dcl}^{eq}(\emptyset )$ . Suppose that $a, b\in \mathcal {M}^{eq}$ are such that . Let $\phi (x, y), \psi (x, z)$ be $\mathcal {L}^{eq}$ -formulas. Then, for an arbitrary $\mathrm {Aut}(M)$ -invariant Keisler measure $\mu : \mathrm {Def}_x(M)\to [0,1]$ ,

$$\begin{align*}\mu(\phi(x, a)\wedge \psi(x, b))\end{align*}$$

only depends on $\mathrm {tp}(a)$ and $\mathrm {tp}(b)$ .

Proof Let be such that $a'\equiv a$ and $b'\equiv b$ . Then, by Corollary 3.8, we get that for any ergodic measure $\nu $ ,

$$\begin{align*}\nu(\phi(x, a)\wedge \psi(x, b))=\nu(\phi(x, a))\nu(\psi(x, b))=\nu(\phi(x, a'))\nu(\psi(x, b'))=\nu(\phi(x, a')\wedge \psi(x, b')).\end{align*}$$

But then, by the ergodic decomposition, Corollary 2.8, we have that for any $\mathrm {Aut}(M)$ -invariant Keisler measure

$$\begin{align*}\mu(\phi(x, a)\wedge \psi(x, b))=\mu(\phi(x, a')\wedge \psi(x, b')).\\[-32pt] \end{align*}$$

Following Remark 2.6, we always compare the forking and universally measure zero ideals in sufficiently saturated and strongly homogeneous models. As noted in the introduction, from [Reference Chernikov, Hrushovski, Kruckman, Krupiński, Moconja, Pillay and Ramsey3] we know that $F(\emptyset )\subseteq \mathcal {O}(\emptyset )$ . We are interested in studying what happens when $F(\emptyset )=\mathcal {O}(\emptyset )$ in order to find a structure where the two sets differ.

Here, the relation denotes non-forking independence. From Corollary 4.1 we obtain that simple $\omega $ -categorical structures where forking coincides with being universally measure zero satisfy the strong independence theorem over the empty set.

Proof Suppose that there are $a, b, c_0, c_1\in \mathcal {M}^{eq}$ as in Definition 4.3. Let $\phi (x, a)$ and $\psi (x, b)$ isolate $\mathrm {tp}(c_0/a)$ and $\mathrm {tp}(c_1/b)$ . By the existence property of non-forking independence, there is $b'\equiv b$ such that . By Corollary 4.1, for any Keisler measure,

$$\begin{align*}\mu(\phi(x, a)\wedge \psi(x, b))=\mu(\phi(x, a)\wedge \psi(x, b')).\end{align*}$$

By simplicity, $\phi (x, a)\wedge \psi (x, b')$ does not fork over the empty set since the independence theorem holds over $\emptyset $ . Hence, by $F(\emptyset )=\mathcal {O}(\emptyset )$ , $\phi (x, a)\wedge \psi (x, b)$ is not universally measure zero, and so is non-forking over the empty set. This proves the strong independence theorem over the empty set for $\mathcal {M}$ .

Remark 4.5. In general, a simple $\omega $ -categorical theory with $\mathrm {acl}^{eq}(\emptyset )=\mathrm {dcl}^{eq}(\emptyset )$ satisfies the independence theorem over the empty set (see, for example, [Reference Marimon17]). However, the condition of $F(\emptyset )=\mathcal {O}(\emptyset )$ yields a strengthening of the independence theorem, where the “base” of the amalgamation is weakly algebraically independent rather than non-forking independent. In fact, while non-forking independence implies weak algebraic independence, the converse does not hold in general.

Definition 4.6. We say that an $\omega $ -categorical simple structure $\mathcal {M}$ is one-based if given $A, B\subseteq \mathcal {M}^{eq}$ algebraically closed, then

If $\mathcal {M}$ is one-based, for $A\subseteq M$ we have that

In particular, in a one-based structure, satisfying the independence theorem over A is equivalent to satisfying the strong independence theorem over A.

However, there are not one-based simple $\omega $ -categorical structures. The only known example of this are $\omega $ -categorical Hrushovski constructions. For these, to satisfy the strong independence theorem is a genuinely stronger requirement than satisfying the independence theorem.

We conclude this section with some consequences for $MS$ -measurable structures in an $\omega $ -categorical context. For a general introduction to $MS$ -measurable structures we suggest [Reference Elwes, Macpherson, Chatzidakis, Macpherson, Pillay and Wilkie7] or the original article [Reference Macpherson and Steinhorn16]. In [Reference Marimon17], I discuss in detail $\omega $ -categorical $MS$ -measurable structures, and we direct the reader to that article for the relevant definitions and some basic results. The general idea is that $MS$ -measurable structures have an associated dimension-measure function assigning each definable set a dimension and a measure. The dimension-measure function is invariant (and definable), the measure always takes strictly positive values, and the dimension and the measure satisfy Fubini’s theorem. Being $MS$ -measurable is a property of a theory. An important feature proved in [Reference Macpherson and Steinhorn16] is that $\mathcal {M}$ is $MS$ -measurable if and only if $\mathcal {M}^{eq}$ is.

Remark 4.7. The dimension d of the dimension-measure function in an $MS$ -measurable theory induces a natural notion of independence: for tuples $b,c$ , and small A, we write

where for a tuple b and A small,

$$\begin{align*}d(b/A)=\min\{d(\phi(x,a))\vert\phi(x,a)\in\mathrm{tp}(b/A)\}.\end{align*}$$

A folklore result attributed to Ben-Yaacov in [Reference Elwes, Macpherson, Chatzidakis, Macpherson, Pillay and Wilkie7] is that this notion of independence corresponds to non-forking independence. See [Reference Marimon17] for a proof of this. Using this fact, we can prove that in $MS$ -measurable theories forking coincides with being universally measure zero (over any small set).

Proof Suppose there is a formula $\phi (x,b)$ which does not fork over A, but has measure zero for every $\mathrm {Aut}(\mathbb {M}/A)$ -invariant Keisler measure. Let $c\vDash \phi (x,b)$ be such that . Since dimension-independence corresponds to non-forking independence, as noted in Remark 4.7, $d(c/Ab)=d(c/A)$ . Let $\psi (x,a)\in \mathrm {tp}(c/A)$ be such that $d(\psi (x,a))=d(c/A)$ . By $MS$ -measurability [Reference Macpherson and Steinhorn16, Proposition 5.3], we have an induced $\mathrm {Aut}(\mathbb {M}/A)$ -invariant Keisler measure $\mu _\psi $ on $\mathrm {Def}_x(\mathbb {M})$ given by

$$\begin{align*}\mu_\psi(\chi(x,d))= \begin{cases} \frac{\mu(\chi(x,d)\wedge \psi(x,a))}{\mu(\psi(x,a))}, & \text{ for } d(\chi(x,d)\wedge \psi(x,a)))=d(\psi(x,a)),\\ 0, & \text{otherwise}, \end{cases} \end{align*}$$

where $\mu $ is the measure in the dimension-measure. Being universally measure zero (over A), we have that $\mu _\psi (\phi (x,b))=\mu (\phi (x,b)\wedge \psi (x,a))=0$ , which contradicts positivity of the measure.

From Corollary 4.1 we obtain that $\omega $ -categorical $MS$ -measurable structures satisfy the strong independence theorem over the algebraic closures of finite sets. Indeed, we also get the corresponding probabilistic independence statement. Recall from Lemma 3.10 that for $A\subseteq M^{eq}$ the algebraic closure of a finite set, and fixing the variable x, there is $A_0\subset A$ finite such that types in the variable x over A are isolated by $\mathcal {L}^{eq}$ -formulas with parameters from $A_0$ .

Theorem 4.9. Let $\mathcal {M}$ be $\omega $ -categorical and $MS$ -measurable. Then, it satisfies the strong independence theorem over the algebraic closures of finite sets:

Let $A\subseteq M^{eq}$ be the algebraic closure of a finite set, $a_0, a_1, b, c$ tuples from $M^{eq}$ . Suppose $a_0\equiv _A a_1$ , , and that , . Then, there is $a^*$ such that $a^*\equiv _{Ab} a_0$ , $a^*\equiv _{Ac} a_1$ , and . Moreover,

$$\begin{align*}\mu(\mathrm{tp}(a_0/Ab)\cup \mathrm{tp}(a_1/Ac))=\frac{\mu(a_0/Ab)\mu(a_1/Ac)}{\mu(a_0/A)}.\end{align*}$$

Proof In [Reference Marimon17] we proved a version of this statement where (instead of ) simplifying some more general results in [Reference Hrushovski11]. We shall use this and Corollary 4.1 to deduce the theorem. Let $\phi (x, b)$ and $\psi (x, c)$ isolate $\mathrm {tp}(a_0/Ab)$ and $\mathrm {tp}(a_1/Ac)$ respectively. Let $\chi (x)$ isolate $\mathrm {tp}(a_i/A)$ , and let $\mu _\chi $ be the $\mathrm {Aut}(M^{eq}/A)$ invariant Keisler measure induced by $\chi (x)$ . We consider $\mu _\chi (\phi (x, b)\wedge \psi (x, c))$ . By extension we can find $c'\equiv _A c$ such that . Since non-forking independence implies weak algebraic independence, by Corollary 4.1

(4.1) $$ \begin{align} \mu_\chi(\phi(x, b)\wedge \psi(x, c'))=\mu_\chi(\phi(x, b)\wedge \psi(x, c)). \end{align} $$

But the former has positive measure by the independence theorem over algebraically closed sets (and d-independence being the same as non-forking independence). Hence, $\phi (x, b)\wedge \psi (x, c)$ must have a realisation which is independent from $bc$ over A. From [Reference Marimon17], for we have that

$$\begin{align*}\mu(\phi(x, b)\wedge \psi(x, c'))=\frac{\mu(\phi(x, b))\mu(\psi(x, c))}{\mu(\chi(x))}.\end{align*}$$

However, multiplying both sides of (4.1) by $\mu (\chi (x))$ ,

$$\begin{align*}\mu(\phi(x, b)\wedge \psi(x, c'))=\mu(\phi(x, b)\wedge \psi(x, c)).\end{align*}$$

This yields the desired equation.

5. The counterexample

We are now ready to introduce our example of a simple $\omega $ -categorical structure with $F(\emptyset )\subsetneq \mathcal {O}(\emptyset )$ . It will be an $\omega $ -categorical Hrushovski construction. For details on Hrushovski constructions the reader may refer to [Reference Wagner, Kaye and Macpherson24, Section 6.2.1]. This construction is also used in [Reference Marimon17] as an example of an $\omega $ -categorical supersimple structure which is not $MS$ -measurable, so the reader may refer to that article for details on this particular construction. The basic idea is that (non-trivial) simple $\omega $ -categorical Hrushovski constructions are not one-based. Hence, there are pairs which are weakly algebraic independent, but forking-dependent. This allows us to build Hrushovski constructions which are simple but which do not satisfy the strong independence theorem.

Remark 5.2. The graph $\mathcal {M}$ in Theorem 5.1 is also extremely amenable in the sense of [Reference Hrushovski, Krupiński and Pillay12]. That is, every type in $S(\emptyset )$ extends to an $\mathrm {Aut}(M)$ -invariant type over M. Since invariant types can be considered as invariant Keisler measures taking only the values $0$ and $1$ , this guarantees that there are some invariant Keisler measures on $\mathcal {M}$ in each variable. To see that these structures are extremely amenable, for a finite tuple $\overline {a}$ from M, consider the type $p(\overline {x})$ given by

$$\begin{align*}\bigcup_{B\subset M \text{ finite}}\left\{\mathrm{tp}(\overline{a}'/B) \mid \overline{a}'\equiv\overline{a}, \mathrm{acl}(\overline{a}B)=\mathrm{acl}(\overline{a})\cup\mathrm{acl}(B), \neg E(c,b) \text{ for }c\in\mathrm{acl}(\overline{a}), b\in\mathrm{acl}(B)\right\}.\end{align*}$$

Substantially, $p(\overline {x})$ asserts that $\overline {x}$ has no relation to $\mathcal {M}$ and is weakly algebraically independent from it. From [Reference Marimon17, Section 4] we can conclude that this type is consistent, complete, and invariant by the extension property and because types of finite tuples are determined by the quantifier-free types of their algebraic closures.

It is commonly known that in a structure with weak elimination of imaginaries, for $A, B\subseteq M$ algebraically closed in $\mathcal {M}$ , we have that

$$\begin{align*}\mathrm{acl}^{eq}(A)\cap\mathrm{acl}^{eq}(B)=\mathrm{acl}^{eq}(A\cap B).\end{align*}$$

Hence, for a and b sharing an edge, , but $\mathrm {tp}(a/b)$ forks over $\emptyset $ . Meanwhile, for a and c at distance two from each other, . Hence, for $\mathcal {M}$ to satisfy the strong independence theorem over the empty set, $\mathcal {M}_f$ should include $5$ -cycles. But we have built $\mathcal {M}_f$ to exclude these. This implies the existence of a formula which does not fork over the empty set, but is universally measure zero. We can give this formula explicitly:

Proof Let $a, b\in M$ share an edge. Now, $\phi (x, a)\wedge \phi (x, b)$ is inconsistent since $\mathcal {M}$ avoids $5$ -cycles. Meanwhile, $\phi (x, a)$ does not fork over the empty set since for c at distance two from a. However, since , for an ergodic measure,

$$\begin{align*}0=\mu(\phi(x, a)\wedge \phi(x, b))=\mu(\phi(x, a))\mu(\phi(x, b))=\mu(\phi(x, a))^2,\end{align*}$$

where the second equality follows by Corollary 3.8 and the last equality by transitivity of the action of $\mathrm {Aut}(M)$ . From the calculation above we get that $\mu (\phi (x, a))=0$ for any ergodic measure. By Corollary 2.8, $\phi (x,a)$ is universally measure zero.

Hence, we get the desired counterexample:

Furthermore, by Lemma 4.8, we get a counterexample to the question of Elwes and Macpherson [Reference Elwes, Macpherson, Chatzidakis, Macpherson, Pillay and Wilkie7]:

The above was also answered in [Reference Evans8, Reference Marimon17], with $\omega $ -categorical Hrushovski constructions as counterexamples.