Integrals and Operators - I. E. Segal

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Sommaire:
I. Introduction.- 1.1 General preliminaries.- 1.2 The idea of measure.- 1.3 Integration as a technique in analysis.- 1.4 Limitations on the concept of measure space.- 1.5 Generalized spectral theory and measure spaces.- Exercises.- II. Basic Integrals.- 2.1 Basic measure spaces.- 2.2 The basic Lebesgue-Stieltjes spaces.- Exercises.- 2.3 Integrals of step functions.- Exercises.- 2.4 Products of basic spaces.- 2.5* Coin-tossing space.- Exercises.- 2.6 Infinity in integration theory.- Exercises.- III. Measurable Functions and Their Integrals.- 3.1 The extension problem.- 3.2 Measurability relative to a basic ring.- Exercises.- 3.3 The integral.- Exercises.- 3.4 Development of the integral.- Exercises.- 3.5 Extensions and completions of measure spaces.- Exercises.- 3.6 Multiple integration.- Exercises.- 3.7 Large spaces.- Exercises.- IV. Convergence and Differentiation.- 4.1 Linear spaces of measurable functions.- Exercises.- 4.2 Set functions.- Exercises.- 4.3 Differentiation of set functions.- Exercises.- V. Locally Compact and Euclidean Spaces.- 5.1 Functions on locally compact spaces.- Exercises.- 5.2 Measures in locally compact spaces.- Exercises.- 5.3 Transformation of Lebesgue measure.- Exercises.- 5.4 Set functions and differentiation in euclidean space.- Exercises.- VI. Function Spaces.- 6.1 Linear duality 152 Exercises.- Exercises.- 6.2 Vector-valued functions.- Exercises.- VII. Invariant Integrals.- 7.1 Introduction.- 7.2 Transformation groups.- Exercises.- 7.3 Uniform spaces.- Exercises.- 7.4 The Haar integral.- 7.5 Developments from uniqueness.- Exercises.- 7.6 Function spaces under group action.- Exercises.- VIII. Algebraic Integration Theory.- 8.1 Introduction.- 8.2 Banach algebras and the characterization of function algebras.- Exercises.- 8.3 Introductory features of Hilbert spaces.- Exercises.- 8.4 Integration algebras.- Exercises.- IX. Spectral Analysis in Hilbert Space.- 9.1 Introduction.- 9.2 The structure of maximal Abelian self-adjoint algebras.- Exercises.- X. Group Representations and Unbounded Operators.- 10.1 Representations of locally compact groups.- 10.2 Representations of Abelian groups.- Exercises.- 10.3 Unbounded diagonalizable operators.- Exercises.- 10.4 Abelian harmonic analysis.- Exercises.- XI. Semigroups and Perturbation Theory.- 11.1 Introduction.- 11.2 The Hille-Yosida theorem.- 11.3 Convergence of semigroups.- 11.4 Strong convergence of self-adjoint operators.- 11.5 Rellich-Kato perturbations.- Exercises.- 11.6 Perturbations in a calibrated space.- Exercises.- XII. Operator Rings and Spectral Multiplicity.- 12.1 Introduction.- 12.2 The double-commutor theorem.- Exercises.- 12.3 The structure of abelian rings.- Exercises.- XIII. C*-Algebras and Applications.- 13.1 Introduction.- 13.2 Representations and states.- Exercises.- XIV. The Trace as a Non-Commutative Integral.- 14.1 Introduction.- 14.2 Elementary operators and the trace.- Exercises.- 14.3 Hilbert algebras.- Exercises.- Selected references.