Abstract
Currently published HOL formalizations of measure theory concentrate on the Lebesgue integral and they are restricted to real-valued measures. We lift this restriction by introducing the extended real numbers. We define the Borel σ-algebra for an arbitrary type forming a topological space. Then, we introduce measure spaces with extended real numbers as measure values. After defining the Lebesgue integral and verifying its linearity and monotone convergence property, we prove the Radon-Nikodým theorem (which shows the maturity of our framework). Moreover, we formalize product measures and prove Fubini’s theorem. We define the Lebesgue measure using the gauge integral available in Isabelle’s multivariate analysis. Finally, we relate both integrals and equate the integral on Euclidean spaces with iterated integrals. This work covers most of the first three chapters of Bauer’s measure theory textbook.
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Hölzl, J., Heller, A. (2011). Three Chapters of Measure Theory in Isabelle/HOL. In: van Eekelen, M., Geuvers, H., Schmaltz, J., Wiedijk, F. (eds) Interactive Theorem Proving. ITP 2011. Lecture Notes in Computer Science, vol 6898. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22863-6_12
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DOI: https://doi.org/10.1007/978-3-642-22863-6_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22862-9
Online ISBN: 978-3-642-22863-6
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