An Effect-Theoretic Account of Lebesgue Integration

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.