Fubini Measures: Theory and Extensions

• Fubini measures are abstract measures that extend classical product measures, enabling integration along definable families in varied mathematical settings.
• They are defined by additivity, null and singleton normalization, and the Fubini property, ensuring consistent exchange of integration order.
• Modern research applies Fubini measures in o-minimal geometry, differential algebra, and functional analysis, unifying proofs of iterated integration theorems.

Fubini measures are an abstraction and generalization of classical product measures, designed to encode “Fubini-type” behavior in a wide variety of contexts—ranging from measure-theoretic and functional-analytic to model-theoretic and categorical settings. Their defining feature is that they enable integration or summing along definable families (“fibers”), lifting the classical exchange of order in summation/integration to generalized frameworks. Modern research elaborates Fubini measures in o-minimal geometry, differential algebra, category theory, functional analysis, probability, and logic, providing rigorous foundations for iterated integration, product constructions, and measure extension in both classical and nonclassical environments (Dries, 27 Dec 2025, Andrzejczak, 2018, Lucyshyn-Wright, 2012, Haak et al., 2024).

1. Formal Definition and Foundational Principles

The notion of a Fubini measure originates in structures where sets and maps are definable (in the model-theoretic sense), typically with a distinguished "base" set $C$ stably embedded in a universe $M$. Let $\Def(M)$ be the category of all definable subsets of powers of $M$ with definable morphisms. A Fubini measure is a function $\mu: \Ob(\mathcal{C})\to A$ (where $A$ is a commutative semiring and $\mathcal{C}\subseteq\Def(M)$ is a subcategory closed under unions, taking definable subsets, and products) such that for every morphism $f: X\to Y$:

• Null and singleton normalization: $\mu(\varnothing)=0$, $\mu(\text{singleton})=1$
• Additivity: $\mu(X\cup Y)=\mu(X)+\mu(Y)$ for disjoint $X,Y$
• Fubini property: All fibers $f^{-1}(y)$ are in $\mathcal{C}$, finitely many values $\mu\left(f^{-1}(y)\right)$ occur (finiteness), level sets $Y_a=\{y:\mu(f^{-1}(y))=a\}$ are definable, and the master identity

$\mu(X)=\sum_{a\in A} a\;\mu(Y_a)$

is satisfied.

This axiomatization ensures “good behavior” under definable families, mirroring classical Fubini–Tonelli for integrals but applicable to a diversity of invariants (dimension, Euler characteristic, Morley rank, motivic measures).

2. Extension and Uniqueness Theorems

One major advance is the theorem that every Fubini measure on $\Def(C)$ (definable subsets of the base $C$) extends uniquely to the category $\Def(C)^{\mathrm{f}}$ of sets fiberable over $C$, where fiberability is inductively defined via successive definable maps whose fibers themselves are fiberable at lower steps (Dries, 27 Dec 2025). In $\omega$-saturated contexts, fiberability coincides with co-analyzability relative to $C$. The inductive extension is as follows:

• Build a tower $\mathcal{C}_0 \subseteq \mathcal{C}_1 \subseteq \ldots \subseteq \Def(C)^{\mathrm{f}}$, starting with those sets isomorphic to $C^n$, then iteratively extending to sets fibered over previously included sets.
• For $X$ fibered over $Y$ by $f:X\to Y$ with fibers in $\mathcal{C}_r$, set:

$\mu(X) = \sum_{a\in A} a\cdot \mu(Y_a)$

where $Y_a=\{y\in Y:\mu(f^{-1}(y))=a\}$.

• Uniqueness follows by unraveling the induction and verifying the Fubini axioms at each level.

This guarantees the persistence of Fubini-type decompositions in motivic and model-theoretic settings, yielding a unified theory of invariants on definable sets (Dries, 27 Dec 2025).

3. Fubini–Tonelli for Arbitrary Measures: Measure-Theoretic Generalizations

In classical measure theory, product measures are only uniquely defined on $\sigma$-finite spaces. Andrzejczak's modification (Andrzejczak, 2018) generalizes Halmos' product construction for arbitrary (possibly non-$\sigma$-finite) measures by crucially restricting initial attention to “o-finite parts” (o-rings generated by finite-measure sets):

• Construct $\mathcal{G}_f=\{A\in\mathcal{G}:\mu(A)<\infty\}$, extend to o-ring $\mathcal{G}^\circ$
• Define product measure $\mu^\circ\otimes\nu^\circ$ on $\mathcal{G}^\circ\otimes\mathcal{E}^\circ$ so $$(\mu^\circ\otimes\nu^\circ)(A\times B)=\mu(A)\nu(B)$$ for $A,B$ of finite measure, extend by countable additivity
• For all of $\mathcal{G}\otimes\mathcal{E}$, set

$$(\mu\otimes\nu)(C) = \begin{cases} (\mu^\circ\otimes\nu^\circ)(C) & C\in \mathcal{G}^\circ\otimes\mathcal{E}^\circ \ \infty & C \notin \mathcal{G}^\circ\otimes\mathcal{E}^\circ \end{cases}$$

This yields a uniquely determined, associative product measure compatible with Fubini–Tonelli: for $f\in L^1(\mu\otimes\nu)$, the iterated integrals agree and existence conditions become local to o-finite parts:

$$\int_{X\times Y} f(x,y)\,d(\mu\otimes\nu)(x,y) = \int_X \left(\int_Y f(x,y)\,d\nu(y)\right)\,d\mu(x) = \int_Y \left(\int_X f(x,y)\,d\mu(x)\right)\,d\nu(y)$$

Thus, integrability w.r.t. the product measure requires “living on o-finite” subsets, enforcing that “purely infinite” sets always get infinite measure (Andrzejczak, 2018).

4. Category-Theoretic Formulations and Fubini-Type Laws

Lucyshyn-Wright (Lucyshyn-Wright, 2012) develops a categorical generalization of the Fubini theorem, abstracting the classical interchange of integration orders via commutative monads in symmetric monoidal and enriched category theory. The setup:

• Let $\Conv$ denote the category of convergence spaces and $\ConVect$ convergence vector spaces over a commutative ring $R$
• A strong symmetric monoidal closed adjunction $F\dashv G:\Conv\leftrightarrows\ConVect$ induces a natural distribution monad $D$
• Under suitable completeness and reflexivity conditions, $D$ is commutative: for any $X,Y$

$$\otimes_{X,Y} = \widetilde{\otimes}_{X,Y}: D X\boxtimes D Y \to D(X\boxtimes Y)$$

This equation represents the Fubini interchange law, recovering the classical result for Radon measures on compact Hausdorff spaces as a special case. The categorical approach enables the extension of Fubini-type laws to a wide spectrum of spaces, including convergence spaces, sheaf-toposes, and synthetic differential structures (Lucyshyn-Wright, 2012).

5. Functional-Analytic and Operator-Theoretic Abstract Fubini Theorems

In functional analysis, Fubini measures underpin the interchange of operations in vector-valued function spaces. Haak and Haase (Haak et al., 2024) prove that for a function $f:O\times\Omega\to\mathbb{C}$ holomorphic in $z$ and measurable in $t$ (with $z\mapsto f(z,\cdot)$ bounded in $L_p(\Omega)$), and given a bp-continuous linear functional $\varphi$ on $H^\infty(O)$ and $\mu\in L_q(\Omega)$, the pairings commute:

$$\langle t\mapsto\varphi( f(\cdot,t) ), \mu \rangle = \varphi(z \mapsto \langle f(z, \cdot), \mu\rangle )$$

Hence, integration and holomorphic evaluation can be interchanged under mild boundedness and continuity assumptions, covering applications to functional calculus of operators and vector-valued analytic functions. Extension to $p=\infty$ relies on additional semi-finiteness of the measure (Haak et al., 2024).

6. Non-Product Measures, Disintegration, and Optimal Transport

Salazar (Salazar, 2018) establishes Fubini–Tonelli-type theorems for non-product measures on $X\times Y$ (beyond the scope of traditional product constructions). For a probability measure $\zeta$ on $X\times Y$ with Borel marginals $\mu$, $\nu$, there exists almost everywhere a family of conditional measures $\nu_x$ on $Y$ (and dually $\mu_y$ on $X$) such that

$\zeta(C) = \int_X \nu_x(C_x)\,\mu(dx)$

with $x\mapsto \nu_x(C_x)$ Borel measurable and corresponding identities for functions:

$$\int_{X\times Y} f(x,y)\,\zeta(dx,dy) = \int_X \left( \int_Y f(x,y)\,\nu_x(dy) \right) \mu(dx)$$

The theory underpins measure-theoretic results in disintegration and yields key results in optimal transport—where couplings can be decomposed along conditional fibers, facilitating duality arguments in cost minimization (Salazar, 2018).

7. Game-Theoretic and Combinatorial Formulations

"The Measure Game" (Crone et al., 2019) provides an alternative combinatorial foundation for Fubini's theorem. Through a two-dimensional infinite game on Cantor space, the equivalence

$(\mu\times\mu)(A)=0 \iff \mu(A_x)=0\ \text{for $\mu$-a.e.$x$} \iff \mu(A^y)=0\ \text{for $\mu$-a.e.$y$}$

is established by relating winning strategies to null sets, effectively bypassing technical integration theory via game dynamics. The deterministic framework is especially powerful for non-measure-theoretic settings (such as Borel determinacy and analytic sets), providing uniformity and constructivity via countable combinatorial mechanisms (Crone et al., 2019).

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In summary, Fubini measures unify and extend classical integration and product-measure principles into algebraic, model-theoretic, categorical, functional-analytic, and combinatorial frameworks. Their core properties ensure exchangeability of summing/integration orders via definable families, support rigorous extension theorems, and underpin categorical and operator-theoretic generalizations of the Fubini–Tonelli paradigm. Applications span from semiring-valued invariants in algebraic geometry and differential fields to measure disintegration in probability, optimal transport, and analytic functional calculus.