Alexander BerensteinDarío GarcíaTingxiang Zou2026-03-222026-03-22202510.1017/jsl.2023.75https://doi.org/10.1017/jsl.2023.75https://andeanlibrary.org/handle/123456789/76725Abstract We study H -structures associated with $SU$ -rank 1 measurable structures. We prove that the $SU$ -rank of the expansion is continuous and that it is uniformly definable in terms of the parameters of the formulas. We also introduce notions of dimension and measure for definable sets in the expansion and prove they are uniformly definable in terms of the parameters of the formulas.enMeasure (data warehouse)Dimension (graph theory)MathematicsCombinatoricsPure mathematicsDiscrete mathematicsarticle