## Constructive algebraic integration theory without choice. Dagstuhl proceedings (2005)

Venue: | Mathematics, Algorithms, Proofs, number 05021 in Dagstuhl Seminar Proceedings. Internationales Begegnungs- und Forschungszentrum (IBFI), Schloss Dagstuhl |

Citations: | 8 - 5 self |

### BibTeX

@INPROCEEDINGS{Spitters05constructivealgebraic,

author = {Bas Spitters},

title = {Constructive algebraic integration theory without choice. Dagstuhl proceedings},

booktitle = {Mathematics, Algorithms, Proofs, number 05021 in Dagstuhl Seminar Proceedings. Internationales Begegnungs- und Forschungszentrum (IBFI), Schloss Dagstuhl},

year = {2005}

}

### OpenURL

### Abstract

Abstract. We present a constructive algebraic integration theory. The theory is constructive in the sense of Bishop, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. Since we avoid impredicative methods the results may also be interpreted in Martin-L type theory or in a predicative topos in the sense of Moerdijk and Palmgren. We outline how to develop most of Bishop’s theorems on integration theory that do not mention points explicitly. Coquand’s constructive version of the Stone representation theorem is an important tool in this process. It is also used to give a new proof of Bishop’s spectral theorem.

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Citation Context ... the axiom of countable choice (CAC) is avoided, as was proposed by Richman [19]. However, a predicative topos in the sense of Moerdijk and Palmgren [16], Aczel’s CZF [1] or Martin-L type theory [15] =-=[17]-=- would be a quite suitable foundations for the current article. Richman [18] states that ‘measure theory and the spectral theorem [...] are major challenges for a choiceless development [of constructi... |

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Citation Context ... The present approach is similar to the C*-algebra approach to topology in the sense that they both study ‘functions’ algebraically. The Boolean algebra approach is similar to that in formal topology =-=[20]-=- [10] [12], where one uses the algebraic properties of open sets, instead of using the open sets themselves. Dagstuhl Seminar Proceedings 05021 Mathematics, Algorithms, Proofs http://drops.dagstuhl.de... |

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Citation Context ...op-style mathematics [2], however the axiom of countable choice (CAC) is avoided, as was proposed by Richman [19]. However, a predicative topos in the sense of Moerdijk and Palmgren [16], Aczel’s CZF =-=[1]-=- or Martin-L type theory [15] [17] would be a quite suitable foundations for the current article. Richman [18] states that ‘measure theory and the spectral theorem [...] are major challenges for a cho... |

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Citation Context ...informally in Bishop-style mathematics [2], however the axiom of countable choice (CAC) is avoided, as was proposed by Richman [19]. However, a predicative topos in the sense of Moerdijk and Palmgren =-=[16]-=-, Aczel’s CZF [1] or Martin-L type theory [15] [17] would be a quite suitable foundations for the current article. Richman [18] states that ‘measure theory and the spectral theorem [...] are major cha... |

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Citation Context ...resentation theorem When X is a locally compact space, then C(X) denotes the space of test-functions, that is, the space of functions with a compact support. We present Coquand’s constructive version =-=[7]-=- of the Stone representation theorem, which we then specialize to f-algebras. The formal space mentioned in the theorem is a space in the sense of formal topology [20] [10] or locale theory [12]. We r... |

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Citation Context ...ekind reals one defines the completion of a metric space using locations as is explained by Richman [19] (sec. 5). A similar construction, the ‘flat completion’, can be found in [25] using ideas from =-=[13]-=- and [3]. We repeat Richman’s definition here. We would like to stress that, given a predicative definition of the Dedekind reals, this definition of the completion is predicative in the sense that on... |

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Citation Context ...nt approach is similar to the C*-algebra approach to topology in the sense that they both study ‘functions’ algebraically. The Boolean algebra approach is similar to that in formal topology [20] [10] =-=[12]-=-, where one uses the algebraic properties of open sets, instead of using the open sets themselves. Dagstuhl Seminar Proceedings 05021 Mathematics, Algorithms, Proofs http://drops.dagstuhl.de/opus/voll... |

12 | The fundamental theorem of algebra: a constructive development without choice
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Citation Context ...ng countable choice. Instead of working in a fixed formal system we work informally in Bishop-style mathematics [2], however the axiom of countable choice (CAC) is avoided, as was proposed by Richman =-=[19]-=-. However, a predicative topos in the sense of Moerdijk and Palmgren [16], Aczel’s CZF [1] or Martin-L type theory [15] [17] would be a quite suitable foundations for the current article. Richman [18]... |

12 |
Algebraic integration theory
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Citation Context ... mentioned in the statement of the theorem. 3.1 Segal’s integration algebras Our definition of an integration f-algebra may be compared to a real commutative integration algebra in the sense of Segal =-=[21]-=- [22]. Definition 4. An integration algebra is a real Abelian algebra A together with a linear functional I such that 1. I(a 2 ) ≥ 0 and I(a 2 ) = 0 if and only if a = 0. 2. For all b ∈ A there exists... |

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Citation Context ... ≤ αb,whenever a ≤ α. An element of a of L is said to be bounded if there exists 2 For us, an f-algebra is always commutative. In fact, one can prove that an Archimedean f-algebra is commutative; see =-=[6]-=-. In [9] it is indicated how this proof can be made constructive. 3sq ∈ � such that |a|∧q = |a|, that is |a| ≤ q. Given any f-algebra one can consider it’s sub-algebra of bounded elements, which is ag... |

7 |
Metric Boolean algebras and constructive measure theory
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(Show Context)
Citation Context ...ee approach where one makes the shift from a measure space to a measure algebra — that is a Boolean algebra with a measure — which in the concrete case is an algebra of sets modulo the null sets; see =-=[8]-=-. These two approaches have counterparts in point-free approaches to topology. The present approach is similar to the C*-algebra approach to topology in the sense that they both study ‘functions’ alge... |

7 |
and Bas Spitters. Formal Topology and Constructive Mathematics: the Gelfand and Stone-Yosida Representation Theorems
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Citation Context ...enever a ≤ α. An element of a of L is said to be bounded if there exists 2 For us, an f-algebra is always commutative. In fact, one can prove that an Archimedean f-algebra is commutative; see [6]. In =-=[9]-=- it is indicated how this proof can be made constructive. 3sq ∈ � such that |a|∧q = |a|, that is |a| ≤ q. Given any f-algebra one can consider it’s sub-algebra of bounded elements, which is again an f... |

4 | Constructive algebraic integration theory
- Spitters
(Show Context)
Citation Context ...ty of Borel sets in [8] are more relevant in this context. However, more research is needed before drawing such conclusions. This note is a continuation of the work on algebraic integration theory in =-=[23]-=-. The difference with the treatment in [23] is that we do not use the axiom of countable choice and the we start with a general f-algebra instead of the f-algebra S(A) of simple functions on a Boolean... |

4 |
Localic completion of generalized metric spaces I. submitted for publication, draft available on web at http://www.cs.bham.ac.uk/~sjv, 2003. A Uncountability of the reals We show that the statement ∀x ∈ N → R∃y ∈ R∀n ∈ N∃m ∈ N|xn − y| ≥ 1/m (2) does not h
- Vickers
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Citation Context ...ion 3.6. Using the Dedekind reals one defines the completion of a metric space using locations as is explained by Richman [19] (sec. 5). A similar construction, the ‘flat completion’, can be found in =-=[25]-=- using ideas from [13] and [3]. We repeat Richman’s definition here. We would like to stress that, given a predicative definition of the Dedekind reals, this definition of the completion is predicativ... |

2 | Trace-class operators
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(Show Context)
Citation Context ... is defined as � n (Afn, fn). The value of TrF (A) is independent of the choice of the basis. If the supremum of TrF (A) over the finite dimensional subspaces F of H exists it is denoted by Tr A, cf. =-=[5]-=-. In this paragraph we use the axiom of dependent choice (DC) to construct a basis, this is needed only to construct the state — a generalized integral — so in case the state can be given in advance, ... |

2 | Constructive mathematics without choice
- Richman
- 2005
(Show Context)
Citation Context ...[19]. However, a predicative topos in the sense of Moerdijk and Palmgren [16], Aczel’s CZF [1] or Martin-L type theory [15] [17] would be a quite suitable foundations for the current article. Richman =-=[18]-=- states that ‘measure theory and the spectral theorem [...] are major challenges for a choiceless development [of constructive mathematics]’ and he expects ‘a choiceless development of this theory to ... |

2 |
Integrals and Operators, volume 228 of Grundlehren der mathematische Wissenschaften
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(Show Context)
Citation Context ...ioned in the statement of the theorem. 3.1 Segal’s integration algebras Our definition of an integration f-algebra may be compared to a real commutative integration algebra in the sense of Segal [21] =-=[22]-=-. Definition 4. An integration algebra is a real Abelian algebra A together with a linear functional I such that 1. I(a 2 ) ≥ 0 and I(a 2 ) = 0 if and only if a = 0. 2. For all b ∈ A there exists α ∈ ... |

1 | Adjoints, absolute values and polar decompositions
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(Show Context)
Citation Context ...can be computed. The Riesz representation theorem states that for any normable functional f there exists y ∈ H such that f(x) = 〈x, y〉. A proof not using the axiom of countable choice can be found in =-=[4]-=- (p.3). Let I and J be integrals on L, then I + J is also an integral L. The functional J is called absolutely continuous with respect to I when J is a continuous linear functional on L1(I). It is cal... |

1 |
Intuitionistic type theory, volume 1
- Martin-L
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(Show Context)
Citation Context ...wever the axiom of countable choice (CAC) is avoided, as was proposed by Richman [19]. However, a predicative topos in the sense of Moerdijk and Palmgren [16], Aczel’s CZF [1] or Martin-L type theory =-=[15]-=- [17] would be a quite suitable foundations for the current article. Richman [18] states that ‘measure theory and the spectral theorem [...] are major challenges for a choiceless development [of const... |