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Bishop's measure theory (BMT) is an abstraction of the measure theory of alocally compact metric space $X$, and the use of an informal notion of aset-indexed family of complemented subsets is crucial to its predicativecharacter. The more general Bishop-Cheng measure theory (BCMT) is aconstructive version of the classical Daniell approach to measure andintegration, and highly impredicative, as many of its fundamental notions, suchas the integration space of $p$-integrable functions $L^p$, rely onquantification over proper classes (from the constructive point of view). Inthis paper we introduce the notions of a pre-measure and pre-integration space,a predicative variation of the Bishop-Cheng notion of a measure space and of anintegration space, respectively. Working within Bishop Set Theory (BST), andusing the theory of set-indexed families of complemented subsets andset-indexed families of real-valued partial functions within BST, we apply theimplicit, predicative spirit of BMT to BCMT. As a first example, we present thepre-measure space of complemented detachable subsets of a set $X$ with theDirac-measure, concentrated at a single point. Furthermore, we translate in ourpredicative framework the non-trivial, Bishop-Cheng construction of anintegration space from a given measure space, showing that a pre-measure spaceinduces the pre-integration space of simple functions associated to it.Finally, a predicative construction of the canonically integrable functions$L^1$, as the completion of an integration space, is included.

Pre-measure spaces and pre-integration spaces in predicative Bishop-Cheng measure theory