In measure theory, a familiar representation theorem due to F. Riesz identifies the dual spaceL_p(X,L λ)* withL_q(X,L, ), where 1/p+1/q=1, as long as 1 ≤ p<∞. However,L_∞( ,L,λ)* 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 ofL_∞(X,L,λ) , examining pathologies and paradoxes, and uncovering some surprising consequences. For instance, a necessary and sufficient condition for a bounded sequence inL_∞(X,L,λ) to be weakly convergent, applicable in the one-point compactification of X, is given.

With a clear summary of prerequisites, and illustrated by examples includingL_∞ Rⁿ) and the sequence spacel_∞, this book makes possibly unfamiliar material, some of which may be new, accessible to students and researchers in the mathematical sciences.