The Dual of L∞(X,L,λ), Finitely Additive Measures and Weak Convergence: A Primer (Repost)
The Dual of L∞(X,L,λ), Finitely Additive Measures and Weak Convergence: A Primer by John Toland
English | PDF,EPUB | 2020 | 104 Pages | ISBN : 3030347311 | 12.95 MB
In measure theory, a familiar representation theorem due to F. Riesz identifies the dual space Lp(X,L,λ)* with Lq(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∞(Rn) 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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