We characterize the surjective linear isometries between spaces of essentially bounded functions on compact subsets of the real line of positive Lebesgue measure, equipped with the Alexiewicz norm. We prove a Banach--Stone type representation: every such isometry is induced by a sign and an increasing lipeomorphism between the associated measure intervals, and admits an explicit weighted composition formula. In particular, any two such Alexiewicz-normed spaces are linearly isometric. We further introduce geometric compatibility conditions on the compact sets (fiber and gap compatibility) which characterize when the induced map between the underlying sets is a homeomorphism or a lipeomorphism.
Categories: math.FA