Elements of information theory
Elements of information theory
Topology in PVS: continuous mathematics with applications
Proceedings of the second workshop on Automated formal methods
A HOL theory of euclidean space
TPHOLs'05 Proceedings of the 18th international conference on Theorem Proving in Higher Order Logics
Theorem Proving with the Real Numbers
Theorem Proving with the Real Numbers
Formal lifetime reliability analysis using continuous random variables
WoLLIC'10 Proceedings of the 17th international conference on Logic, language, information and computation
Three chapters of measure theory in Isabelle/HOL
ITP'11 Proceedings of the Second international conference on Interactive theorem proving
Formalization of entropy measures in HOL
ITP'11 Proceedings of the Second international conference on Interactive theorem proving
Formalization of finite-state discrete-time Markov chains in HOL
ATVA'11 Proceedings of the 9th international conference on Automated technology for verification and analysis
A type theory for probability density functions
POPL '12 Proceedings of the 39th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages
Formal analysis of fractional order systems in HOL
Proceedings of the International Conference on Formal Methods in Computer-Aided Design
Formal probabilistic analysis of cyber-physical transportation systems
ICCSA'12 Proceedings of the 12th international conference on Computational Science and Its Applications - Volume Part III
Formalization of Measure Theory and Lebesgue Integration for Probabilistic Analysis in HOL
ACM Transactions on Embedded Computing Systems (TECS) - Special Issue on Modeling and Verification of Discrete Event Systems
Quantitative analysis of information flow using theorem proving
ICFEM'12 Proceedings of the 14th international conference on Formal Engineering Methods: formal methods and software engineering
Probabilistic Relational Reasoning for Differential Privacy
ACM Transactions on Programming Languages and Systems (TOPLAS)
Formal reasoning about classified markov chains in HOL
ITP'13 Proceedings of the 4th international conference on Interactive Theorem Proving
An approach for lifetime reliability analysis using theorem proving
Journal of Computer and System Sciences
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Lebesgue integration is a fundamental concept in many mathematical theories, such as real analysis, probability and information theory. Reported higher-order-logic formalizations of the Lebesgue integral either do not include, or have a limited support for the Borel algebra, which is the canonical sigma algebra used on any metric space over which the Lebesgue integral is defined. In this paper, we overcome this limitation by presenting a formalization of the Borel sigma algebra that can be used on any metric space, such as the complex numbers or the n-dimensional Euclidean space. Building on top of this framework, we have been able to prove some key Lebesgue integral properties, like its linearity and monotone convergence. Furthermore, we present the formalization of the “almost everywhere” relation and prove that the Lebesgue integral does not distinguish between functions which differ on a null set as well as other important results based on this concept. As applications, we present the verification of Markov and Chebyshev inequalities and the Weak Law of Large Numbers theorem.