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The Dual of L (X,L, ), Finitely Additive Measures and Weak Convergence
Anglais
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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 p<. However, 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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Anglais
