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

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(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.

Autorentext

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).

Inhalt
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.