Constructive algebraic integration theory without choice

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-LAƒÂ¶f 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.

[1]  F. Richman.  The fundamental theorem of algebra: a constructive development without choice , 2000 .

[2]  Irving Segal,et al.  Algebraic Integration Theory , 1965 .

[3]  I. Moerdijk,et al.  Sheaves in geometry and logic: a first introduction to topos theory , 1992 .

[4]  S. Lane,et al.  Sheaves In Geometry And Logic , 1992 .

[5]  Bengt Nordström,et al.  Programming in Martin-Lo¨f's type theory: an introduction , 1990 .

[6]  Bas Spitters,et al.  Constructive algebraic integration theory , 2006, Ann. Pure Appl. Log..

[7]  Thierry Coquand,et al.  Metric Boolean algebras and constructive measure theory , 2002, Arch. Math. Log..

[8]  D. Fremlin,et al.  Topological Riesz Spaces and Measure Theory , 1974 .

[9]  Erik Palmgren,et al.  Type theories, toposes and constructive set theory: predicative aspects of AST , 2002, Ann. Pure Appl. Log..

[10]  F. William Lawvere,et al.  Metric spaces, generalized logic, and closed categories , 1973 .

[11]  T. Coquand.  About Stone's notion of spectrum , 2005 .

[12]  Fred Richman,et al.  Constructive Mathematics without Choice , 2001 .

[13]  P. Aczel,et al.  Notes on constructive set theory , 1997 .

[14]  A. Troelstra,et al.  Constructivism in Mathematics: An Introduction , 1988 .

[15]  Steven Vickers,et al.  Localic completion of generalized metric spaces I , 2005 .

[16]  Marcello M. Bonsangue,et al.  Generalized Metric Spaces: Completion, Topology, and Powerdomains via the Yoneda Embedding , 1995, Theor. Comput. Sci..

[17]  G. Buskes,et al.  Almost f-algebras: Commutativity and the Cauchy-Schwarz Inequality , 2000 .

[18]  G. Sambin.  Intuitionistic Formal Spaces — A First Communication , 1987 .

[19]  Per Martin-Löf,et al.  Intuitionistic type theory , 1984, Studies in proof theory.

[20]  Martin Schechter.  16 – Trace Class Operators , 1981 .

[21]  Thierry Coquand,et al.  Formal Topology and Constructive Mathematics: the Gelfand and Stone-Yosida Representation Theorems , 2005, J. Univers. Comput. Sci..

[22]  Douglas S. Bridges,et al.  ADJOINTS, ABSOLUTE VALUES AND POLAR DECOMPOSITIONS , 1997 .