In measure theory, a familiar representation theorem due to F. Riesz identifies the dual space L p(X, L,λ)* with L q(X, L,λ), where 1/p+1/q=1, as long as 1 ≤ p<∞. However, L ∞(X, 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 of L ∞(X, L,λ)*, examining pathologies and paradoxes, and uncovering some surprising consequences. For instance, a necessary and sufficient condition for a bounded sequence in L ∞(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 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.
