The Dual of L¿(X, L,¿), Finitely Additive Measures and Weak Convergence

Zusatztext

<p>In measure theory, a familiar representation theorem due to F. Riesz identifies the dual space<i>L</i><sub>p</sub>(X,L,)* with<i>L</i><sub>q</sub>(X,L,), where 1/p+1/q=1, as long as 1 p<. However,<i>L</i><sub></sub>(X,L,)* cannot be similarly described, and is instead represented as a class of finitely additive measures.</p><p>This book provides a reasonably elementary account of the representation theory of<i>L</i><sub></sub>(X,L,)*, examining pathologies and paradoxes, and uncovering some surprising consequences. For instance, a necessary and sufficient condition for a bounded sequence in<i>L</i><sub></sub>(X,L,) to be weakly convergent, applicable in the one-point compactification of X, is given.</p><p>With a clear summary of prerequisites, and illustrated by examples including<i>L</i><sub></sub>(<b>R</b><sup>n</sup>) and the sequence space<i>l</i><sub></sub>, this book makes possibly unfamiliar material, some of which may be new, accessible to students and researchers in the mathematical sciences.</p>

Autorenportrait

John Toland FRS is a mathematical analyst who worked in nonlinear partial differential equations and served as Director of the Isaac Newton Institute for Mathematical Sciences in Cambridge (2011-2016). He was awarded the London Mathematical Society Berwick Prize (2000) and the Royal Society Sylvester Medal (2012).

Weitere Details