A classic of pure mathematics, this advanced graduate-level text explores the intersection of functional analysis and analytic function theory. Close in spirit to abstract harmonic analysis, it is confined to Banach spaces of analytic functions in the unit disc. The author devotes the first four chapters to proofs of classical theorems on boundary values and boundary integral representations of analytic functions in the unit disc, including generalizations to Dirichlet algebras. The fifth chapter contains the factorization theory of Hp functions, a discussion of some partial extensions of the factorization, and a brief description of the classical approach to the theorems of the first five chapters. The remainder of the book addresses the structure of various Banach spaces and Banach algebras of analytic functions in the unit disc. Enhanced with 100 challenging exercises, a bibliography, and an index, this text belongs in the libraries of students, professional mathematicians, as well as anyone interested in a rigorous, high-level treatment of this topic.
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If X is a set, the collection of all subsets of X forms a ring, using the operations
A σ-ring of subsets of X is a subring of the ring of all subsets of X which is closed under the formation of countable unions (and, a fortiori, closed under the formation of countable intersections).
Suppose that X is a locally compact Hausdorff topological space, e.g., n-dimensional Euclidean space or a closed subset thereof. The Baire subsets of X are the members of the smallest σ-ring of subsets of X which contains every compact Gδ, i.e., every compact subset of X which is the intersection of a countable number of open sets. The Borel subsets of X are the members of the smallest σ-ring of subsets of X which contains every compact set. In Euclidean space, every compact (closed and bounded) set is a Gδ; hence, if X is a closed subset of Euclidean space, the Baire and Borel subsets of X coincide. When X is the real line or a closed interval on the line, the ring of Baire (Borel) subsets of X may also be described as the σ-ring generated by the half-open intervals [a, b).
If X is a locally compact Hausdorff space, a positive Baire (Borel) measure on X is a function μ which assigns to every Baire (Borel) subset of X a non-negative real number (or +∞), in such a way that
whenever A1, A2,… is a sequence of pairwise disjoint Baire (Borel) sets in X. The Borel measure μ is called regular if for each
orel set A
the infimum being taken over the open sets U containing A. A Baire measure is always regular, and each Baire measure has a unique extension to a regular Borel measure. For this reason (and others) we shall discuss only Baire measures on X.
The positive Baire measure μ is called finite if μ(A) is finite for each Baire set A. If X is compact, μ is finite if and only if μ(X) is finite.
Suppose X is the real line or a closed interval. Let F be a monotone increasing (non-decreasing) function on X which is continuous from the left:
Define a function μ on semi-closed intervals [a, b) by
Then μ has a unique extension to a positive Baire measure on X. The measure μ is fin...
