Keisler Measures in Model Theory

• Keisler measures are finitely additive probability measures on definable sets, bridging classical type theory with probabilistic analysis in model theory.
• They support core concepts like invariance, definability, and generic stability, which are essential in both NIP and stable theoretical frameworks.
• Applied in group theory and exchangeability, Keisler measures enable operations such as Morley products and convolution to analyze model-theoretic dynamics.

Keisler measures are probability measures on definable sets in first-order structures. In the standard formulation, a Keisler measure over a parameter set $A$ in variables $x$ is a finitely additive probability measure on the Boolean algebra of $A$-definable subsets of the $x$-sort; equivalently, it is a regular Borel probability measure on the Stone space $S_x(A)$. Complete types embed as Dirac measures, so the theory of types appears as the $\{0,1\}$-valued fragment of the theory of Keisler measures. This perspective has become central in contemporary model theory because it supports probabilistic analogues of invariance, definability, Morley products, convolution on definable groups, and exchangeable sampling constructions (Ackerman et al., 20 Oct 2025, Khanaki, 2021, Hoffmann, 6 Apr 2025).

1. Definitions, representations, and local variants

The basic dictionary between syntax and topology is fundamental. If $\mu$ is a Keisler measure over $A$, then for each $A$-formula $\varphi(x)$, the value $x$0 is the measure of the corresponding clopen subset of $x$1. Conversely, regular Borel probability measures on $x$2 restrict to finitely additive probability measures on definable sets. Global Keisler measures are measures over a monster model, and an $x$3-invariant measure on $x$4 may be regarded equivalently as an automorphism-invariant finitely additive probability measure on definable sets over $x$5 (Braunfeld et al., 2024, Marimon, 2022).

A local version restricts attention to a fixed formula $x$6. Then one works with the Boolean algebra $x$7 generated by instances of $x$8, the local type space $x$9, and local Keisler measures on that algebra. This localization is useful in NIP and in stability-in-a-model arguments, where global pathologies can be avoided while preserving the relevant combinatorics of a single formula or a finite Boolean closure of formulas (Gannon, 2018, d'Elbée et al., 1 Jan 2026).

In continuous logic, the same notion appears with essentially the same formal role: a Keisler measure on $A$0 is a regular Borel probability measure on the metric type space, equivalently a positive linear functional on $A$1. Recent work shows that this continuous version preserves the central model-theoretic uses of the discrete theory, including Morley products, smoothness, generic stability, and distal regularity (Anderson, 2023).

2. Invariance, definability, and tame regularity conditions

Several regularity notions organize the theory. A global measure $A$2 is $A$3-invariant if $A$4 depends only on $A$5. For an invariant measure and a formula $A$6, one obtains a function

$A$7

defined on $A$8. If every such function is Borel, $A$9 is Borel-definable over $x$0; if every such function is continuous, $x$1 is definable over $x$2. Finite satisfiability means that every definable set of positive measure has a realization in the base model. Smoothness is stronger: $x$3 is smooth when it has a unique global extension, or equivalently when its restriction to a base model extends uniquely to larger models (1009.3566, Conant et al., 2021, Anderson, 2023).

In NIP theories, generically stable measures are characterized by definability together with finite satisfiability, and this is equivalent to symmetry of Morley products and total indiscernibility of the Morley sequence. The same picture persists locally: for an NIP formula $x$4, a local measure is generically stable over $x$5 if and only if it is finitely approximable over $x$6 by empirical averages of points from $x$7 (1009.3566, Gannon, 2018). A related sequential viewpoint shows that, in countable NIP theories, finitely satisfiable measures over countable models are limits of average measures on finite tuples from the model, and that finite approximation is equivalent to definability plus sequential approximation (Gannon, 2021).

Outside NIP, the literature distinguishes several nonequivalent tameness notions. The frequency interpretation measure condition, or fim, is the measure-theoretic analogue of generic stability and is emphasized as the correct replacement beyond NIP. In arbitrary theories, fim measures are closed under convex combinations and commute with all Borel-definable measures, while definable plus finitely satisfiable measures need not have comparable behavior (Chernikov et al., 2024, Conant et al., 2021). A broader measure-theoretic regularity notion is dependence: a measure is dependent when the corresponding family of definable functions is a Glivenko–Cantelli class relative to that measure. All measures in NIP theories are dependent; all types and all fim measures are dependent in arbitrary theories (Khanaki, 2021).

3. Morley products, convolution, and semigroup structure

The Morley product is the basic binary operation on measures. If $x$8 is invariant and Borel-definable over $x$9, and $S_x(A)$0 is another measure, then

$S_x(A)$1

for suitable $S_x(A)$2. Iterating this yields the Morley powers $S_x(A)$3 and the countable Morley power $S_x(A)$4, which underlies model-theoretic sampling constructions (Ackerman et al., 20 Oct 2025, 1009.3566).

The existence and formal behavior of Morley products depend sharply on tameness hypotheses. Over countable parameter sets in countable theories, Borel-definable measures are closed under Morley products and satisfy associativity. Over uncountable parameter sets, both closure and associativity can fail: there are Borel-definable types $S_x(A)$5 such that $S_x(A)$6 is not Borel-definable, and there are examples where $S_x(A)$7. If one factor is definable, however, associativity is restored in the general framework developed there (Conant et al., 2021).

For groups, Morley products induce definable convolution. If $S_x(A)$8 is a definable or type-definable group and $S_x(A)$9 are measures on $\{0,1\}$0, then

$\{0,1\}$1

whenever the product is defined. In NIP groups, the spaces of invariant or finitely satisfiable measures become compact left-continuous semigroups under this operation, extending Newelski’s semigroup product on types. The push-forward map to the compact group $\{0,1\}$2 is a continuous semigroup homomorphism from definable convolution to classical convolution, linking model-theoretic dynamics to the Kawada–Ito/Wendel description of idempotent measures on compact groups (Chernikov et al., 2020, Chernikov et al., 2022).

A different but related convolution framework arises for actions of $\{0,1\}$3. On general topological groups, ordinary Borel probability measures may not admit a well-behaved convolution because $\{0,1\}$4 in general. The introduction of $\{0,1\}$5-additive Borel probability measures resolves this for $\{0,1\}$6: on compact type spaces, $\{0,1\}$7-additive measures coincide with the usual regular Borel probability measures, so the classical notion of Keisler measure is recovered while convolution with measures on $\{0,1\}$8 becomes well defined (Hoffmann, 6 Apr 2025).

4. Classification theorems and structural results

Some of the strongest results concern idempotent measures. In stable groups, a global Keisler measure $\{0,1\}$9 is idempotent if and only if it is the unique right-invariant, equivalently unique left-invariant, measure on the type-definable subgroup $\mu$0. This is the stable-group analogue of the classical theorem identifying idempotent probability measures on compact groups with Haar measures on compact subgroups (Chernikov et al., 2020).

Recent work extends this classification in a definable direction. For a definable or type-definable group $\mu$1, the study of idempotent measures under convolution isolates generic transitivity as the structural property permitting stable-like group theory around generically stable objects. In the abelian case, if $\mu$2 is abelian and $\mu$3 is fim and idempotent, then $\mu$4 is the unique left- and right-invariant measure on the type-definable subgroup $\mu$5, and $\mu$6 is the smallest type-definable subgroup supporting $\mu$7. In NIP, this yields a bijective correspondence between generically stable idempotent measures and type-definable fsg subgroups. The same work proves generic transitivity for idempotent generically stable types in stable groups, abelian groups, inp-minimal groups, and rosy theories, and uses Keisler’s randomization theory to transfer type-theoretic arguments to measures (Chernikov et al., 2024).

Local stability produces an even more rigid picture. If $\mu$8 is stable in a model $\mu$9, then every local $A$0-measure is a weighted sum of at most countably many types: $A$1 In this setting there are no strongly continuous local measures, local definability is automatic, and the Morley product is commutative. The corresponding evaluation map on pairs of local measures satisfies the double-limit property, lifting the classical stability criterion from types to local measures (d'Elbée et al., 1 Jan 2026).

In NIP groups, minimal left ideals in measure convolution semigroups exhibit additional rigidity. The ideal subgroups of minimal left ideals are always trivial, and in the definably amenable case the unique minimal left ideal of invariant measures is a Bauer simplex whose extreme points come from right $A$2-generic types. For countable NIP groups, the revised Newelski conjecture is proved: the $A$3-topology on the ideal group $A$4 is Hausdorff, and this yields a canonical Haar measure on $A$5 and an explicit construction of a minimal left ideal in the measure semigroup (Chernikov et al., 2022, Chernikov et al., 2024).

Ultraproducts supply another structural regime. Over a countable ultraproduct with countably incomplete ultrafilter, finitely approximable measures are exactly pseudo-finite measures, meaning ultralimits of finite convex combinations of realized measures. In NIP ultraproducts one therefore has

$A$6

and for definable pseudo-finite measures the Morley product agrees with the pseudo-finite product coming from finite-level approximants (Gannon, 2023).

5. Exchangeability, sampling, and broader applications

Keisler measures now interact substantially with exchangeability theory. A global Borel-definable measure $A$7 over the countable universal homogeneous $A$8-uniform hypergraph gives rise, through its countably iterated Morley product $A$9, to a probability measure on countable labeled structures by push-forward. For measures $A$0 built from Borel hypergraphons $A$1, this generic sampling exactly reproduces standard hypergraphon sampling: $A$2 Hence every ergodic $A$3-invariant measure on countable $A$4-uniform hypergraphs is the pushforward of the countably iterated Morley product of a global Borel-definable Keisler measure; for $A$5, this yields a representation theorem for ergodic exchangeable graphs over a monster model of the Rado graph (Ackerman et al., 20 Oct 2025).

A complementary line of work studies when automorphism-invariance already forces exchangeability. For many countable homogeneous structures, invariant Keisler measures can be represented as invariant random expansions of the base structure. Under overlap-closedness hypotheses, every invariant random expansion by a suitably low-arity hereditary class is exchangeable, and therefore invariant Keisler measures become exchangeable as well. This gives explicit descriptions of invariant Keisler measure spaces for random hypergraphs, generic $A$6-free hypergraphs, tetrahedron-free $A$7-hypergraphs, $A$8-petal-free $A$9-hypergraphs, and parity hypergraphs (Braunfeld et al., 2024).

Arithmetic applications also exist. For bounded perfect PAC fields and perfect Frobenius fields, recent constructions produce Keisler measures on definable subsets of varieties by reducing formulas to test formulas, encoding them via finite Galois extensions, and assigning probabilities through a Markov chain on intermediate fields. As a consequence, every group definable in such fields is definably amenable (Chatzidakis et al., 18 Apr 2025).

In continuous logic, the NIP theory of Keisler measures extends the classical equivalences among fim, fam, definability plus approximate realization, self-commutation of the Morley product, and total indiscernibility of the iterated Morley product. Distality is characterized by the statement that every generically stable measure is smooth, or equivalently that all pairs of generically stable measures are weakly orthogonal. These facts yield analytic versions of distal regularity, cutting lemmas, and strong Erdős–Hajnal phenomena (Anderson, 2023).

6. Limitations, counterexamples, and boundary phenomena

The modern theory is shaped as much by counterexamples as by classification theorems. One central lesson is that idempotence alone is insufficient for subgroup-style classification: the abelian classification requires fim or generic stability, and the general theory explicitly notes the existence of idempotent measures that are not tame enough to be controlled by invariant subgroup data (Chernikov et al., 2024).

Morley products also behave badly outside tame settings. Over uncountable parameter sets, Borel-definable measures need not be closed under Morley products, and associativity may fail. Outside NIP, the notions dfs, fam, and fim diverge: there are complete global types that are definable and finitely satisfiable over every small model but not finitely approximated over any small model, and there are fam types that are not fim. Likewise, definable measures need not commute with dfs types in arbitrary theories (Conant et al., 2021).

The relation between forking and invariant measures is also subtler than might be expected. In an NIP theory there are formulas that do not fork over $\varphi(x)$0 but have measure $\varphi(x)$1 under every global $\varphi(x)$2-invariant Keisler measure; this pathology disappears if the theory is also first-order amenable, in which case every nonforking formula has positive measure under some global automorphism-invariant measure (Pillay et al., 2023). In simple and $\varphi(x)$3-categorical simple theories, the gap can be sharper: there are nonforking formulas that are universally measure zero, and there are simple theories with definable groups that are not definably amenable, although definable groups in small theories are definably amenable (Chernikov et al., 2021, Marimon, 2022).

These limitations clarify the present conceptual landscape. Keisler measures generalize types, but they do not merely interpolate between types and classical probability. Their behavior depends delicately on invariance, definability, dependence, NIP, local stability, and group-theoretic or combinatorial structure. The cumulative effect of the recent literature is a stratified theory: rigid and classification-friendly in stable, NIP, distal, or fim settings; exchangeability-compatible in several homogeneous contexts; and demonstrably wild outside those regimes (Khanaki, 2021, Braunfeld et al., 2024).