apply measure theory

Traditional risk measures apply only to single random variables, limiting their use in systemic and set-valued risk assessment. Method: This paper introduces **Set-Valued Risk Measures (SRMs)**—real-valued mappings defined on nonempty, closed, bounded, and almost-surely bounded sets of random variables—and develops a tailored axiomatic framework compatible with set operations. Leveraging convex analysis, set-valued functions, the (L^infty) space, and regular finitely additive measures on the unit ball, it establishes a **dual representation theorem for convex SRMs**, fully characterizing them via regular finite additivity. It further defines worst-case SRMs to address systemic risk evaluation and ambiguity- or robustness-aware decision-making. Contribution/Results: The proposed framework provides a rigorous, axiomatically complete foundation for systemic risk measurement and robust portfolio optimization, bridging theoretical depth with practical flexibility in financial applications.

Classical probabilistic constructions—such as almost-invariant σ-algebras, ergodic decompositions, the de Finetti theorem, and the zero–one law—lack a unified structural explanation within standard probability theory. Method: We construct a category whose objects are probability spaces and whose morphisms are measure-preserving Markov kernels, identified up to almost-sure equality. Our approach integrates categorical methods (notably the Markov category and its dagger structure), standard Borel space theory, and null-isomorphism techniques. Contribution/Results: We provide the first structural, categorical proof of the ergodic decomposition theorem; characterize almost-invariant σ-algebras uniformly as both limits and colimits; establish a dual limit–colimit characterization of invariant structures in random dynamical systems; and unify three foundational limit theorems—the de Finetti theorem, the zero–one law, and the ergodic decomposition—within a single abstract categorical framework. This advances the structural coherence, universality, and intrinsic categorical nature of probability theory.

In uncertainty quantification, existing risk measures face a tension between overly restrictive law invariance and insufficient probabilistic sufficiency. Method: We introduce the novel concept of “partial law invariance” to unify these two properties. We formally define partial and strong partial law invariance; establish a new theoretical bridge between Kusuoka representations and real-world uncertainty; and propose families of partially law-invariant risk measures—namely, expected shortfall and entropy-based measures. Contribution/Results: We derive necessary and sufficient conditions for compatibility of such risk measures and provide computationally tractable optimization formulations. Numerical experiments demonstrate that the proposed measures exhibit superior modeling flexibility and robustness under heterogeneous uncertainty. This work extends the foundational theory of risk measurement and furnishes new analytical tools for financial risk management and behavioral decision modeling.

The weak analogy between Shannon entropy and signed measures, coupled with the lack of geometric characterization for information sets, hinders a structural understanding of information. Method: We propose the Logarithmic Decomposition (LD) framework, which represents the information structure of random variables as definable “logarithmic atoms” over the sample space Ω. By extending Yeung’s I-measure, integrating signed measure theory with information geometry, and introducing the notion of logarithmic decomposability, the framework enables structured criteria for positive and negative entropy atoms. Contribution/Results: LD geometrically reconstructs common information and sufficient statistics; strictly distinguishes dyadic from triadic systems—beyond the capability of I-measure; unifies set-theoretic interpretations of mutual information, conditional entropy, and related quantities; establishes a foundation for quality-oriented information theory; and admits a natural extension to continuous distributions.

Existing probabilistic programming semantics struggle to model random functions whose codomains are conical structures—such as continuous data types—and lack rigorous mathematical foundations for sampling primitives. Method: We develop a novel integration theory on the category of measurable cones: first adapting the Pettis integral to cone-valued functions, and introducing two exponential comonads—one based on stable measurable functions and another on integrable analytic functions over cones. Integrating measurable cone theory, linear logic models, and semantics for probabilistic PCF, we construct a fully abstract cone-based model. Contribution/Results: This yields the first sound and adequate denotational model supporting continuous sampling primitives. It provides a unified, computationally meaningful semantic foundation for both call-by-value and call-by-push-value probabilistic languages, overcoming fundamental limitations of traditional discrete or Euclidean frameworks in modeling randomized programs.

This work addresses the challenge of verifying mathematical proofs generated by large language models by formally encoding, for the first time, an entire advanced undergraduate probability textbook—including its measure-theoretic foundations—into Lean. To bridge the semantic gap between the textbook’s exposition and the abstract formalism of the Mathlib library, the authors introduce an “interface lemma” strategy. Combined with structured proof engineering and formalization techniques specific to measure theory, this approach yields a reusable, machine-verifiable infrastructure spanning fourteen textbook chapters. The resulting formalization not only provides rigorous verification of all stated theorems and explicit articulation of their assumptions but also establishes a robust foundation for reliable AI-assisted mathematics, educational applications, and future formalization efforts in probability theory.

This study investigates robust risk measures derived from worst-case convex risk measures over uncertainty sets, aiming to characterize their continuity conditions and uncover the duality relationship between such risk measures and the underlying uncertainty sets. Drawing on tools from convex analysis, duality theory, and set-valued analysis, the work proposes two complementary dual frameworks—each relying on distinct geometric assumptions—to establish a unified set-valued dual representation. The main contributions include precise criteria for the continuity of robust risk measures and dual characterizations of both the risk measures and the uncertainty sets, thereby systematically elucidating the deep interplay between them.

This work addresses the limitations of existing formalisms for hyperproperties in capturing quantitative aspects inherent in real-world systems, such as numerical relationships in information flow control. To overcome this, the paper introduces Quantitative Hyper-Logic (QHL), a novel framework that reformulates hyperproperty specifications using measure theory, replacing classical Boolean quantifiers with measures to support nested quantitative structures. Leveraging Hoeffding’s inequality and extreme value theory, the authors develop an efficient statistical verification algorithm and provide rigorous analyses of sample complexity and statistical guarantees. Experimental evaluation on quantitative information-flow benchmarks demonstrates that QHL substantially outperforms conventional qualitative approaches, offering superior expressiveness and verification capabilities that better align with the demands of practical systems.

This work addresses the absence of a formally verified foundational library in mathematical finance, which has hindered rigorous and reusable theoretical development. Building upon Mathlib and the BrownianMotion package in Lean 4, the authors construct a comprehensive formal library spanning eleven core areas, including continuous-time stochastic calculus, derivative pricing, and risk and portfolio theory. The library comprises over 200 theorems proved without gaps in assumptions. Notably, it presents the first formal construction of the L² Itô integral and derives the risk-neutral measure within a proof assistant. Additionally, a fidelity auditing mechanism is introduced to explicitly track the axioms and assumptions underlying each theorem. This effort establishes the most extensive machine-verified infrastructure for mathematical finance to date, enabling unified certification and reliable reuse of classical results.

This work addresses the limitations of traditional Gaussian assumptions in accurately representing complex uncertainties, which often lead to information loss and reduced accuracy in multi-stage measurement and control processes. To overcome these challenges, the paper proposes a scalable precision framework based on Gaussian Mixture Models (GMMs), leveraging GMMs as universal approximators of probability density functions. The approach integrates closed-form uncertainty propagation algorithms with memory-efficient computational strategies, thereby transcending the representational constraints of Gaussian methods while maintaining computational tractability. Experimental evaluations in manufacturing and metrology scenarios—such as circular factories—demonstrate that the proposed method significantly enhances the fidelity of uncertainty characterization and propagation, outperforming conventional Gaussian-based techniques.