Isometries and geometric liftings for Alexiewicz-normed $L^{\mathrm{\infty }}$ spaces

We study spaces of essentially bounded functions on compact subsets of the real line, equipped with the Alexiewicz norm given by the supremum norm of the primitive. Using the associated measure projection, we classify their surjective linear isometries as weighted composition operators determined by a sign and an increasing bi-Lipschitz map between the corresponding measure intervals. We also give geometric criteria characterizing when this interval-level map lifts to a homeomorphism or to a bi-Lipschitz homeomorphism between the underlying compact sets.