Algebraic Integration Theory
There are a number of important analytical situations in which the approach to an integral through a measure space is unnatural or technically disadvantageous. In fact, we have developed a major part of the theory from the point of view of integration lattices, following the fundamental ideas of Daniell. However, in the examples of integration lattices considered so far, e.g., the real step functions on a basic measure space or the continuous real-valued functions of compact support on a locally compact space, there is another inherent element of structure which in many respects is more important; these particular lattices are also algebras, and there are definite advantages from a broad viewpoint to a formulation of integration theory which starts from a linear functional on an algebra rather than a lattice. This approach to the theory of integration, which might be called the algebraic approach, is not restricted to function algebras alone, arises naturally from the foundations of probability theory, and is explicitly or implicitly indicated in a variety of other situations, e.g., commutative spectral theory in Hilbert space, the theory of integration in infinite-dimensional linear spaces, harmonic analysis on Abelian and more general groups (in particular, the L2 theory), and developments closely paralleling integration theory in the theory of rings of operators.
KeywordsHilbert Space Measure Space Compact Space Banach Algebra Hermitian Form
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