1 Introduction.- 2 Notation and Preliminaries.- 3 L8 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 L8(X,L,?).- 9 L8* when X is a Topological Space.- 10 Reconciling Representations.- References.- Index.
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<8. However, L8(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 L8(X,L,?)*, examining pathologies and paradoxes, and uncovering some surprising consequences. For instance, a necessary and sufficient condition for a bounded sequence in L8(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 L8(Rn) and the sequence space l8, this book makes possibly unfamiliar material, some of which may be new, accessible to students and researchers in the mathematical sciences.
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).
Collana: SpringerBriefs in Mathematics
Dimensioni: 235 x 155 mm Ø 186 gr
Illustration Notes:1 Illustrations, black and white
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