An effect-theoretic account of Lebesgue integration. (English) Zbl 1352.81011
Ghica, Dan (ed.), Proceedings of the 31st conference on the mathematical foundations of programming semantics (MFPS XXXI), Nijmegen, The Netherlands,
June 22–25, 2015. Amsterdam: Elsevier. Electronic Notes in Theoretical Computer Science 319, 239-253, electronic only (2015).
Summary: Effect algebras have been introduced in the 1990s in the study of the foundations of quantum mechanics, as part of a quantum-theoretic version of probability theory. This paper is part of that programme and gives a systematic account of Lebesgue integration for \([0,1]\)-valued functions in terms of effect algebras and effect modules. The starting point is the ‘indicator’ function for a measurable subset. It gives a homomorphism from the effect algebra of measurable subsets to the effect module of \([0,1]\)-valued measurable functions which preserves countable joins. It is shown that the indicator is free among these maps: any such homomorphism from the effect algebra of measurable subsets can be thought of as a generalised probability measure and can be extended uniquely to a homomorphism from the effect module of \([0,1]\)-valued measurable functions which preserves joins of countable chains. The extension is the Lebesgue integral associated to this probability measure. The preservation of joins by it is the monotone convergence theorem.
For the entire collection see [Zbl 1342.68016].
For the entire collection see [Zbl 1342.68016].
MSC:
| 81P10 | Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) |
| 03G12 | Quantum logic |
| 26A42 | Integrals of Riemann, Stieltjes and Lebesgue type |
| 28B15 | Set functions, measures and integrals with values in ordered spaces |
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