λ-Variation and Hausdorff Distance

✍ Scribed by Franciszek Prus-Wiśniowski

• Book ID: 102943410
• Publisher: John Wiley and Sons
• Year: 2006
• Tongue: English
• Weight: 646 KB
• Volume: 158
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19921580120

✦ Synopsis

Abstract

It is shown that points of varying monotonicity of a function play an important role in calculation of its λ‐variation. It is proven that for a given continuous function x: [0,1] → IR of bounded λ‐variation, the function A → λ‐var(x, A) is continuous on the metric space (ℋ︁, d~H~) of all closed subsets of [0, 1] with Hausdorff distance. Another function from (ℋ︁, d~H~) into ℛ is used to characterize the compactness in CΛBV~c~, i.e. in the Banach space of continuous functions that are continuous in λ‐variation. As a corollary the separability of CΛABV~c~ is proven.