The Dual of L (X,L, ), Finitely Additive Measures and Weak Convergence: A Primer

John Toland - Collection Yellow Sale 2023

99 pages, parution le 06/02/2020

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Résumé

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.

In measure theory, a familiar representation theorem due to F. Riesz identifies the dual space L _p (X,L, )* with L _q (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 ( R ⁿ ) 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.

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.John Toland FRS is a mathematical analyst who worked in nonlinear partial differential equations and served as Director of the Isaac Newton Institute for Mathematical Sciences in Cambridge (2011-2016). He was awarded the London Mathematical Society Berwick Prize (2000) and the Royal Society Sylvester Medal (2012).