John Toland 
The Dual of L∞(X, L, λ), Finitely Additive Measures and Weak Convergence [PDF ebook] 
A Primer

In measure theory, a familiar representation theorem due to F. Riesz identifies the dual space Lp(X, L, λ)* with Lq(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.

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