The Dual of L∞(X,L,λ), Finitely Additive Measures and Weak Convergence : A Primer
著者
書誌事項
The Dual of L∞(X,L,λ), Finitely Additive Measures and Weak Convergence : A Primer
（SpringerBriefs in mathematics）
Springer, c2020
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注記
Includes bibliographical references (p. 95-96) and index
内容説明・目次
内容説明
In measure theory, a familiar representation theorem due to F. Riesz identifies the dual space Lp(X,L, )* with Lq(X,L, ), where 1/p+1/q=1, as long as 1 p< . However, L (X,L, )* cannot be similarly described, and is instead represented as a class of finitely additive measures.
This book provides a reasonably elementary account of the representation theory of L (X,L, )*, examining pathologies and paradoxes, and uncovering some surprising consequences. For instance, a necessary and sufficient condition for a bounded sequence in L (X,L, ) to be weakly convergent, applicable in the one-point compactification of X, is given.
With a clear summary of prerequisites, and illustrated by examples including L (Rn) and the sequence space l , this book makes possibly unfamiliar material, some of which may be new, accessible to students and researchers in the mathematical sciences.
目次
1 Introduction.- 2 Notation and Preliminaries.- 3 L and its Dual.- 4 Finitely Additive Measures.- 5 G: 0-1 Finitely Additive Measures.- 6 Integration and Finitely Additive Measures.- 7 Topology on G.- 8 Weak Convergence in L (X,L, ).- 9 L * when X is a Topological Space.- 10 Reconciling Representations.- References.- Index.
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