The Dual of L8(X,L,?), Finitely Additive Measures and Weak Convergence

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.