On the Formalization of the Lebesgue Integration Theory in HOL

Abstract

Lebesgue integration is a fundamental concept in many mathematical theories, such as real analysis, probability and information theories. 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 report, 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.

Proof Pearl: A Probabilistic Proof for the Girth-Chromatic Number Theorem

“…Obviously some amount of probability theory. Different fragments of probability theory are now formalized in many theorem provers, including HOL4, HOL-light, PVS, Mizar and Isabelle [12,17,18,21,23]. Surprisingly, for the proof presented here, not much more than Markov's Inequality is required.…”

Evaluation of anonymity and confidentiality protocols using theorem proving

“…The integral used in the definition of product measure is the Lebesgue integral which we formalized in [18].…”

“…We also used the formalization of Lebesgue integral [18] to formalize the main statistical properties of random variables, such as the expectation and the variance. Further details about this formalization can be found in [17].…”

“…In particular, this paper presents an extension of existing theories of measure, Lebesgue integration and probability [18] to cater for measures involving multiple random variables. Building upon this formalization, we present a higher-order-logic formalization of the Kullback-Leibler (KL) divergence [6] from which we can derive the formalization of most of the information leakage measures presented in the literature so far and the information leakage degrees that we propose in this paper.…”

“…We build upon the existing formalization of measure, integration and probability [18], and information theories [19] to provide a complete framework to formally reason about quantitative properties of information flow analysis within the sound core of the HOL4 theorem prover and thus guarantee accuracy of the analysis. In particular, this paper presents an extension of existing theories of measure, Lebesgue integration and probability [18] to cater for measures involving multiple random variables.…”