The Dual of L8(X,L,?), Finitely Additive Measures and Weak Convergence A Primer

In measure theory, a familiar representation theorem due to F. Riesz identifies the dual space L_p(X,L,?)* with L_q(X,L,?), where 1/p+1/q=1, as long as 1 = p<8. However, L₈(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 L₈(X,L,?)*, examining pathologies and paradoxes, and uncovering some surprising consequences. For instance, a necessary and sufficient condition for a bounded sequence in L₈(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 L₈(Rⁿ) and the sequence space l₈, this book makes possibly unfamiliar material, some of which may be new, accessible to students and researchers in the mathematical sciences.