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.

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).