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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2207.08684 |
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| _version_ | 1866909338970357760 |
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| author | Petrakis, Iosif Zeuner, Max |
| author_facet | Petrakis, Iosif Zeuner, Max |
| contents | Bishop's measure theory (BMT) is an abstraction of the measure theory of a locally compact metric space $X$, and the use of an informal notion of a set-indexed family of complemented subsets is crucial to its predicative character. The more general Bishop-Cheng measure theory (BCMT) is a constructive version of the classical Daniell approach to measure and integration, and highly impredicative, as many of its fundamental notions, such as the integration space of $p$-integrable functions $L^p$, rely on quantification over proper classes (from the constructive point of view). In this 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 an integration space, respectively. Working within Bishop Set Theory (BST), and using the theory of set-indexed families of complemented subsets and set-indexed families of real-valued partial functions within BST, we apply the implicit, predicative spirit of BMT to BCMT. As a first example, we present the pre-measure space of complemented detachable subsets of a set $X$ with the Dirac-measure, concentrated at a single point. Furthermore, we translate in our predicative framework the non-trivial, Bishop-Cheng construction of an integration space from a given measure space, showing that a pre-measure space induces 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_08684 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Pre-measure spaces and pre-integration spaces in predicative Bishop-Cheng measure theory Petrakis, Iosif Zeuner, Max Logic Functional Analysis Bishop's measure theory (BMT) is an abstraction of the measure theory of a locally compact metric space $X$, and the use of an informal notion of a set-indexed family of complemented subsets is crucial to its predicative character. The more general Bishop-Cheng measure theory (BCMT) is a constructive version of the classical Daniell approach to measure and integration, and highly impredicative, as many of its fundamental notions, such as the integration space of $p$-integrable functions $L^p$, rely on quantification over proper classes (from the constructive point of view). In this 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 an integration space, respectively. Working within Bishop Set Theory (BST), and using the theory of set-indexed families of complemented subsets and set-indexed families of real-valued partial functions within BST, we apply the implicit, predicative spirit of BMT to BCMT. As a first example, we present the pre-measure space of complemented detachable subsets of a set $X$ with the Dirac-measure, concentrated at a single point. Furthermore, we translate in our predicative framework the non-trivial, Bishop-Cheng construction of an integration space from a given measure space, showing that a pre-measure space induces 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. |
| title | Pre-measure spaces and pre-integration spaces in predicative Bishop-Cheng measure theory |
| topic | Logic Functional Analysis |
| url | https://arxiv.org/abs/2207.08684 |