The Dual of L∞(X,L,λ), Finitely Additive Measures and Weak Convergence: A Primer

(SpringerBriefs in Mathematics) 1st ed. 2020 Edition 

by John Toland (Author) 

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_∞(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.