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Algebraic Theory of Measure and Integration: Second English Edition
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AMS Chelsea Publishing
1963; 378 pp; hardcover
Volume: 161
ISBN-10: 0-8218-5273-6
ISBN-13: 978-0-8218-5273-6
List Price: US$53 Member Price: US$47.70
Order Code: CHEL/161.H

By generalizing the concept of point function to that of a function ("soma" function) over a Boolean ring, Carathéodory gives in this book an elegant algebraic treatment of measure and integration.

• Somas: 1.1-2 The axiomatic method; 1.3-7 Elementary theory of somas; 1.8-13 Somas as elements of a Boolean algebra; 1.14-16 The main properties of the union; 1.17-22 The decomposability of somas; 1.23-24 The intersection of an infinite number of somas; 1.25-32 Limits and bounds
• Sets of Somas: 2.33-40 Sets of somas closed under a binary operation; 2.41-46 Complete rings; 2.47-53 Ordinal numbers of the second class; 2.54-55 Hereditary sets of somas; 2.56-64 Homomorphisms of rings of somas
• Place Functions: 3.65-68 Finitely-valued place functions; 3.69-75 Nests of somas; 3.76-79 Altering the domain of definition; 3.80-88 Principal properties of the soma functions $$\alpha(X)$$ and $$\beta(X)$$
• Calculation with Place Functions: 4.89-94 Limit processes; 4.95-106 Elementary operations on place functions; 4.107-110 Uniform and absolute convergence; 4.111-117 Composition of place functions; 4.118-125 Homomorphisms of place functions
• Measure Functions: 5.126-128 Additive and union-bounded soma functions; 5.129-130 Measurability; 5.131-135 Measure functions; 5.136-140 The measure function on its ring of measurability; 5.141-143 Sequences of measure functions and their limits; 5.144-147 Transformation of measure functions by homomorphisms; 5.148-153 The Borel-Lebesgue content
• The Integral: 6.154 Fields of place functions; Measurable place functions; 6.155-162 The notion of the integral; 6.163-166 Linearity of the integral and the integration of place functions of arbitrary sign; 6.167-172 Comparable measure functions and the Lebesgue decomposition; 6.173-175 Abstract differentials; 6.176-177 The absolute continuity of two comparable measure functions; 6.178-180 Transformation of the integral by means of homomorphisms
• Application of the Theory of Integration to Limit Processes: 7.181-183 The theorem of Egoroff; 7.184-189 Continuity of the integral as a functional; 7.190-197 Convergence in the mean; 7.198-205 Ergodic theory
• The Computation of Measure Functions: 8.206-210 Maximal measure functions; 8.211-215 The bases of an arbitrary measure function; 8.216-221 Relative measurability
• Regular Measure Functions: 9.222-224 The definition and principal properties of regular measure functions; 9.225-229 Inner measure; 9.230-235 Comparison of inner and outer measures; 9.236-240 The arithmetic mean of the inner and outer measures
• Isotypic Regular Measure Functions: 10.241-244 The principal properties of isotypic measure functions; 10.245-248 The Jordan decomposition of completely additive soma functions; 10.249-255 The difference of two isotypic regular measure functions; 10.256-257 Comparable outer measures
• Content Functions: 11.258-259 The definition of content functions; 11.260-267 Reduced content functions and their homomorphisms; 11.268-271 The Jessen infinite-dimensional torus; 11.272-278 The Vitali covering theorem; 11.279-282 The Lebesgue integral; 11.283-284 Comparable content functions; 11.285-289 Linear measure
• Appendix: Somas as elements of partially ordered sets: 12.290-297 A new axiom system for somas; 12.298-302 The partitioning of a set into classes; 12.303-304 Partially ordered sets; 12.305-308 Applications to the theory of somas; 12.309-312 Systems of somas that are not isomorphic to systems of subsets of a set
• Bibliography: Earlier publications by Constantin Carathéodory on the algebraization of measure and integral
• List of symbols
• Index