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

## A Primer

Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

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Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

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*_{∞}(**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.

Riesz Representation Finitely additive measures Weak convergence Yosida-Hewitt Essential range Extreme points

- DOI https://doi.org/10.1007/978-3-030-34732-1
- Copyright Information The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
- Publisher Name Springer, Cham
- eBook Packages Mathematics and Statistics
- Print ISBN 978-3-030-34731-4
- Online ISBN 978-3-030-34732-1
- Series Print ISSN 2191-8198
- Series Online ISSN 2191-8201
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