Random Set Theory

• Random Set Theory is a framework for modeling uncertainty where outcomes are inherently set-valued, ensuring measurability via Borel mappings and measurable selections.
• It employs capacity, belief, and the Aumann expectation functionals to extend classic Bayesian and Dempster–Shafer methods for quantifying uncertainty.
• Its applications range from PDEs and multi-agent systems to risk measures, using computational techniques that guarantee convergence and tractability.

Random set theory provides a unifying mathematical framework for modeling, quantifying, and analyzing uncertainty where outcomes are fundamentally set-valued, encompassing standard probabilistic, interval, and imprecise or fuzzy uncertainty models as special cases. A random set is, in essence, a measurable set-valued mapping from a probability space into a collection of closed subsets—often in a Euclidean, Banach, or more general Polish space—endowed with a suitable $\sigma$-algebra. This theory underpins a range of quantification methodologies in uncertainty analysis, statistical inference, combinatorics, and engineering, and is central to the modern analysis of data fusion, large-scale multi-agent systems, spectral theory of random operators, and set-valued risk measures.

1. Foundational Definitions and Measurability

Random set theory begins with a measurable space $(\Omega, \Sigma, P)$, and a separable, complete metric or normed space $E$ with its Borel $\sigma$-algebra $\mathcal{B}(E)$. A set-valued map $X:\Omega\to 2^E$ is called a random set if for every $\omega\in\Omega$, the value $X(\omega)$ is nonempty, closed, and for each Borel (or equivalently, closed or open) set $B\subset E$, the hit event $X^-(B) = \{\omega : X(\omega)\cap B\neq\emptyset\}$ is $\Sigma$-measurable (Karakašević et al., 2021, Taraldsen, 2019).

The Castaing–Valadier Fundamental Measurability Theorem states this is equivalent to the existence of a sequence of measurable selections $\{x_k\}$ with $x_k(\omega)\in X(\omega)$, whose images are dense in the random set almost surely. Standard set operations—finite unions, intersections, closures, convex hulls—preserve the measurability of random sets, permitting the construction of complex derived random sets (Karakašević et al., 2021, Taraldsen, 2019).

2. Descriptors: Capacity, Belief, Plausibility, and Aumann Expectation

A key tool is the capacity functional (hitting functional): for a random set $X$ and Borel $B\subset E$,

$T_X(B) = P\{\omega: X(\omega)\cap B\ne\emptyset\}.$

$T_X$ is monotone, normalized ($T_X(\emptyset)=0$, $T_X(E)=1$), and continuous from below, but typically non-additive. The belief functional $$\operatorname{Bel}_X(B) = P\{ X(\omega)\subset B \}$$ offers a lower probability, and satisfies $\operatorname{Bel}_X(B) + T_X(B^c) = 1$, unifying frameworks such as Dempster–Shafer evidence theory, possibility measures, and p-boxes (Karakašević et al., 2021, Hoang et al., 2018).

The Aumann expectation generalizes conventional expectation to set-valued outputs. For a random set $X$ taking closed, bounded values in a Banach space and admitting integrable selections: $$\mathbb{E}[X] = \operatorname{cl}\Bigl\{ \int_\Omega x(\omega) dP(\omega) : x(\cdot) \text{ a measurable selection of } X \Bigr\}.$$ If $X$ is convex and compact a.s., $\mathbb{E}[X]$ is likewise convex and compact. For interval-valued sets in $\mathbb{R}$, this reduces to taking the interval $$[\mathbb{E}[\underline x], \mathbb{E}[\overline x]]$$ (Karakašević et al., 2021).

3. Operations and Theoretical Properties

Random set theory extends classical operations to the set-valued domain:

• Minkowski sum: $(X+Y)(\omega) = X(\omega)\oplus Y(\omega)$; expectation is additive.
• Intersection: $(X\cap Y)(\omega)$, with capacity bounded above by the minima of the factors’ capacities.
• Convex hull: $\operatorname{conv} X(\omega)$, which is measurable, and expectation commutes with convexification.
• Scalar multiplication and translated sets are random sets with correspondingly transformed capacity functionals (Karakašević et al., 2021, Hoang et al., 2018).

Random finite set (RFS) theory generalizes to the case where both the individual elements and the cardinality (possibly zero) are random, with major applications in multi-object estimation, tracking, and decentralized control. RFSs naturally obviate explicit labeling or association problems and are amenable to Poisson and Gaussian mixture approximations in engineering (Doerr et al., 2018, Doerr et al., 2020).

4. Inference, Approximation, and Computational Aspects

Inference with random sets generalizes Bayesian and Dempster–Shafer frameworks. The inverse problem—updating a prior random set with observation-induced random sets—proceeds via intersection of sets and an updated probability law on $\Omega$, extending Dempster’s rule. The capacity transform density (plausibility transform) $\pi_T(x) \propto P(X \ni x)$ provides a pivotal bridge: updating $\pi_T(x)$ follows a standard Bayesian procedure, enabling the use of MCMC and similar methods for sampling and approximate inference (Hoang et al., 2018).

Random discrete-set approximations reduce computational complexity in high-dimensional or structurally complex random set problems by focusing on sampled subsets whose membership and set-theoretic operations are tractable to evaluate. Convergence can be quantified in terms of Hausdorff distance and mean-squared error on support functions (Hoang et al., 2018, Lew et al., 2020).

Computational strategies in applied domains include polynomial chaos expansions, stochastic response surfaces, importance sampling over credal sets, and nested Monte Carlo approaches, often targeting efficient estimation of lower/upper distributions and envelope bounds for quantities of interest (Karakašević et al., 2021).

5. Applications: PDEs, Multi-Agent Systems, and Reachability

Partial Differential Equations: Random set theory rigorously characterizes uncertainty propagation and output envelopes in elliptic and hyperbolic PDEs where coefficients or input parameters are uncertain in a set-valued sense. The pointwise solution domains can be represented as interval-valued random sets, with belief and plausibility quantified in terms of capacity functionals. This unifies interval, probabilistic, and credal uncertainty quantification and permits explicit bounds on solution behavior (Karakašević et al., 2021).

Control of Multi-Agent and Swarm Systems: RFS theory is now standard in the modeling and control of large collaborative swarms, as the state of the swarm is naturally a finite set with random cardinality and agent locations. Centralized and decentralized RFS control laws are derived through Gaussian mixture approximations, PHD filters, and optimal control tools such as ILQR and sparsity-promoting LQR, which can interpolate smoothly between fully centralized and fully decentralized architectures while maintaining stability and near-optimality (Doerr et al., 2018, Doerr et al., 2020).

Reachability Analysis: For high-dimensional nonlinear and stochastic systems, sampling-based approaches justified by random set theory construct approximations to reachable sets via convex hulls of propagated sample paths, guaranteeing almost sure convergence in Hausdorff metric and enabling rigorous over-approximation. Adversarial sampling, which targets coverage of underrepresented boundary regions, further accelerates convergence (Lew et al., 2020).

Spectral Theory of Random Operators: The spectrum and eigenvalue set of a random operator, defined via measurable graphs in Hilbert space, are themselves random closed sets. The measurability of these spectra leverages the fundamental theorems for measurable selections and projections, enabling unified treatment across bounded and unbounded, self-adjoint and nonselfadjoint cases (Taraldsen, 2019).

6. Random Sets in Combinatorics, Learning Theory, and Additive Number Theory

The probabilistic method in combinatorics and learning theory leverages random set theory to analyze combinatorial and phase-transition phenomena:

• VC dimension and shattering: Random families of sets exhibit zero-one laws for shattering thresholds that can be precisely located for given $t$-subset coverage, interval patterns, and permutations. The minimal number of random sets to achieve full shattering of all $t$-sets is sharply determined by moment and concentration inequalities (Godbole et al., 2013, Segal-Halevi et al., 2016).
• Concentration inequalities: The UI (Uniform Imbalance) dimension refines VC dimension concepts, controlling tail bounds for the imbalance between empirical and expected frequencies in random-set-selected subpopulations (Segal-Halevi et al., 2016).
• Additive combinatorics: Studying the sum and difference sets of random and correlated pairs of finite sets (with parameterized dependency structure) reveals limiting probabilities, continuity properties, and phase transitions for phenomena such as sum-dominance and minimal more-sums-than-differences (MSTD) set construction (Do et al., 2014).

7. Set-Valued Functionals and Risk Measures

Random set theory underpins modern set-valued risk measures, distributional functionals, and depth regions. The support function formalism enables the extension of scalar gauge functionals (expectation, quantiles, expectiles, AVaR) to set-valued and conditional set-valued functionals. These include:

• Generalized conditional Aumann expectation (for $g = \mathbb{E}$),
• Depth-trimmed regions and Tukey half-space depth (for $g=q_\alpha$),
• Cone-quantiles and set-valued risk envelopes,
• With strong results on measurability, law-determination, monotonicity, positive homogeneity, and translation equivariance in both unconditional and conditional forms (Fissler et al., 18 Apr 2025).

Dual representations express these set-valued functionals as intersections of half-spaces or as support functions of super-level sets, and measurable-version theorems guarantee the existence and uniqueness of maximal random closed convex sets representing the functional's output.

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In sum, random set theory provides rigorous, unifying principles and methodologies across probability, statistical inference, spectral theory, control, combinatorics, and risk analysis, centering on the interplay between set-valued measurability, functional representation, and operational closure. It yields both abstract foundational results and computationally effective methods for heterogeneous uncertainty quantification and complex stochastic modeling (Karakašević et al., 2021, Hoang et al., 2018, Fissler et al., 18 Apr 2025, Taraldsen, 2019).