Regular Conditional Distributions of Correspondences

• The topic defines regular conditional distributions of correspondences as a generalization of classical conditional distributions to set-valued and random elements.
• It establishes key regularity properties—convexity, closedness, and compactness—through nowhere equivalence and extension of positive linear functionals.
• The framework is vital for applications in stochastic geometry, Bayesian games, and random closed sets, ensuring existence, uniqueness, and invariance in probabilistic models.

A regular conditional distribution of correspondences is a measure-theoretic and functional-analytic framework for describing distributions of random elements, sets, or processes where only partial information—such as prescribed expectations, moments, or marginal distributions of selected functionals—is specified. These objects generalize standard regular conditional distributions to set-valued correspondences and permit the analysis of existence, uniqueness, and structural properties (including symmetry and invariance) under minimal regularity constraints. The notion is fundamental in areas such as stochastic geometry, probabilistic modeling, Bayesian games, and mathematical economics, and is tightly connected to the theory of extensions of positive linear functionals, regularity moduli, and the interplay between measure-theoretic properties (notably, nowhere equivalence) and regularity features like convexity, compactness, and closed graph preservation.

1. Foundational Concepts

Central to the theory are measurable correspondences $F: T \to \mathcal{P}(X)$ (assigning to almost every $t$ a nonempty subset $F(t) \subset X$), sub-$\sigma$-algebras $\mathcal{F}$, and $\mathcal{T}$, and regular conditional distributions (RCDs) arising via measurable selections $f: T \to X$. For each such $f$, the RCD $\mu^{f|\mathcal{F}}$ is defined, satisfying:

• For every $t \in T$, $\mu^{f|\mathcal{F}}(t, \cdot)$ is a Borel probability measure on $X$;
• For all Borel sets $B \subset X$, $t \mapsto \mu^{f|\mathcal{F}}(t, B)$ equals $\mathcal{F}$-conditional expectation of $\mathbf{1}_{\{f \in B\}}$.

Associated sets of distributions include: $$D_F^{\mathcal{T}} = \left\{ \lambda \circ f^{-1}: f \text{ is a } \mathcal{T}\text{-measurable selection of } F \right\}$$ and analogously, the set of RCDs given $\mathcal{F}$: $$\mathcal{R}_F^{T,\mathcal{F}} = \left\{ \mu^{f|\mathcal{F}}: f \text{ is a } \mathcal{T}\text{-measurable selection of } F \right\}$$ These definitions generalize the notion of conditional expectation and probability kernel to the context of correspondences and set-valued maps.

2. Regularity Properties and Extension Theory

The core technical apparatus leverages the extension of positive linear functionals $\Phi$ defined on subspaces of functions $\mathcal{G}$ over a state-space $\mathcal{E}$ to larger vector lattices $\mathcal{E}$, preserving positivity and upper semi-continuity (regularity). The fundamental extension result (a Kantorovich-type theorem) states: if $\mathcal{G}$ majorizes $\mathcal{E}$ (i.e., $|v| \leq g$ for $v \in \mathcal{E}$, $g \in \mathcal{G}$), every positive $\Phi$ on $\mathcal{G}$ admits a positive extension to $\mathcal{E}$.

Regularity is further encoded via a "regularity modulus" $\chi: \mathcal{E} \rightarrow [0,\infty]$, a lower semi-continuous function such that for all $g \in \mathcal{G}$, the sublevel sets $$\mathcal{H}_g = \{X \in \mathcal{E}: \chi(X) \leq g(X)\}$$ are relatively compact. The main existence theorem asserts the equivalence between the feasibility of a random element $\xi$ realizing the prescribed moments $\Phi(\mathfrak{g})$, with a bound $E[\chi(\xi)] \leq r$, and the regularity condition: $$\sup\left\{ \Phi(\mathfrak{g}): \mathfrak{g} \in \mathcal{G}, \mathfrak{g} \leq \chi \right\} \leq r$$ This decouples the realisability problem into positivity and regularity parts.

3. The Role of Nowhere Equivalence

A pivotal measure-theoretic condition underlying the regularity properties of RCDs of correspondences is nowhere equivalence between $\mathcal{T}$ and $\mathcal{F}$. Formally, $\mathcal{T}$ is nowhere equivalent to $\mathcal{F}$ if every $D \in \mathcal{T}$ of positive measure contains a $\mathcal{T}$-measurable subset $D_0$ such that for all $\mathcal{F}$-measurable $D_1 \subset D$, $\lambda(D_0 \triangle D_1) > 0$. This feature is necessary and sufficient for convexity, closedness, compactness, and closed graph preservation of $D_F^{\mathcal{T}}$ and $\mathcal{R}_F^{T,\mathcal{F}}$, or for the purification property (i.e., existence of deterministic selections matching prescribed conditional distributions) (Otsuka, 19 Sep 2025).

Older results required $\mathcal{F}$ to be countably generated and often imposed atomlessness (no atoms) on the base space; the nowhere equivalence framework removes these technical assumptions and yields a fully general characterization of the measure-theoretic underpinning of regularity.

4. Applications to Point Processes and Random Closed Sets

This machinery finds concrete implementation in stochastic geometry, notably in constructing point processes with given correlation measures or random closed sets with prescribed two-point covering or contact distributions (Lachieze-Rey et al., 2011):

• For point processes $\xi$ on locally compact spaces, quadratic polynomial functionals $$\mathfrak{g}_h(Y) = \sum_{x_i, x_j \in Y, i \neq j} h(x_i, x_j)$$ encode second-order structure. Existence of $\xi$ with $E[\mathfrak{g}_h(\xi)] = \Phi(\mathfrak{g}_h)$ and finiteness/local-exclusion properties (hard-core) is guaranteed by regularity moduli of the form

$$\chi_\varepsilon(Y) = \sum_{x,y \in Y, x \neq y} \|x - y\|^{-(d+\varepsilon)}$$

or more generally $$\chi^{hc}_\psi(Y) = \sum_{x,y \in Y, x \neq y} \psi(d(x,y))$$ for monotone $\psi$ blowing up at $0$.

• For random closed sets $\xi$, realization of finite-dimensional indicators (such as $p_{x,y} = P\{x,y \in \xi\}$) requires upper semi-continuity and regularity moduli ensuring that sublevel sets retain compactness, leading to closed realizations.

Verification typically relies on approximating $\chi$ via the test functions and separating the combinatorial positivity (e.g., positive-definiteness of kernels) from the analytic control of small-scale behavior.

5. Invariance and Symmetry Properties

Invariance under group actions plays a fundamental role. When both the function space $\mathcal{G}$ and the linear functional $\Phi$ are invariant under a group $\Theta$ of transformations (e.g., translations or symmetry operations), the extension process ensures that the constructed random elements $\xi$ will also possess stationarity or invariance (distributional invariance under $\Theta$) (Lachieze-Rey et al., 2011). Formally, if for all $\theta \in \Theta$, $\Phi(\theta \mathfrak{g}) = \Phi(\mathfrak{g})$, then $\xi$ can be chosen so that $\theta\xi$ has the same law as $\xi$.

Such symmetry constraints are crucial in the paper of spatial processes, random fields, and models in physics, image analysis, and econometrics, where invariance properties encode physical or economic homogeneity.

6. Connections to Bayesian Games and Large Game Equilibria

Regular conditional distributions of correspondences underpin existence results in economic models, especially Bayesian games and large games. In Bayesian games, regularity and convexity of conditional expectation correspondences with respect to inter-player informational structures are both necessary and sufficient for existence of pure-strategy equilibria (He et al., 2013): $$\text{Game has a pure strategy equilibrium} \iff \forall i,\; T_i \text{ has no }G_i\text{-atom}$$ For large games with general trait or type spaces (possibly non-countably generated), the equivalence between nowhere equivalence and regularity yields existence of pure-strategy Nash equilibria and supports the purification principle (Otsuka, 19 Sep 2025, He et al., 2021), even when payoffs depend on the joint distribution over agents and actions ("semi-anonymous settings"). This broadens the mathematical economics toolkit for models with heterogeneous agents and aggregate externalities.

7. Advanced Themes and Mathematical Formulations

In application contexts, such as the description of random fields by systems of conditional distributions (Khachatryan, 2022), the specification and reconstruction of large-scale structure from finitely-parameterized conditional laws depend on factorization and quasilocality properties, and lead to explicit necessary and sufficient conditions for the unique existence of random field distributions.

Mathematical tools central to the theory include formulas for disintegration and reconstruction, e.g.,

$$E[\mathfrak{g}(\xi)] = \Phi(\mathfrak{g}), \quad \sup\{\Phi(\mathfrak{g}): \mathfrak{g} \leq \chi \} \leq r$$

and structural assertions about compactness, convexity, and upper semi-continuity. In operator algebra, analogous principles are seen in "regular" C*-correspondences, where regularity of module structures supports well-behaved decomposition and equivalence relations (Bilich et al., 8 Nov 2024).

Table: Core Regularity Properties and Their Measure-Theoretic Equivalence

Property	Required for Regularity	Measure-Theoretic Condition
Convexity	Closed-valued correspondence	Nowhere equivalence of σ-algebras
Closedness	Closed-valued correspondence	Nowhere equivalence
Compactness	Compact-valued correspondence	Nowhere equivalence
Closed graph	Parameterized generators	Nowhere equivalence
Purification	Matching conditional laws	Nowhere equivalence

These equivalences enable generalization to non-countably generated settings.

Summary

Regular conditional distributions of correspondences provide a comprehensive framework for the analysis, construction, and structural understanding of random elements, sets, and fields under partial specification. The equivalence between regularity properties (convexity, closedness, compactness, preservation of closed graph) and measure-theoretic conditions like nowhere equivalence assures robust and general existence results, supports purification arguments in large games, and enables structured control over symmetry and invariance. Applications encompass stochastic geometry, games with large populations, conditional independence modeling, and even operator algebraic settings, marking the framework as foundational in advanced probability, analysis, and mathematical economics.