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 L (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 L (X, L, )*, examining pathologies and paradoxes, and uncovering some surprising consequences. For instance, a necessary and sufficient condition for a bounded ...
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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 L (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 L (X, L, )*, examining pathologies and paradoxes, and uncovering some surprising consequences. For instance, a necessary and sufficient condition for a bounded sequence in L (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 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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