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We study the asymptotic behaviour of suitably defined seminorms in generalmetric measure spaces. As a particular case we provide new and shorter proofsof the Maz'ya-Shaposhnikova's theorem on the asymptotic behaviour of thefractional Sobolev $s$-seminorm, in the setting of metric measure spaces andwith general mollifiers, as well as of the Ludwig's result on finitedimensional Banach spaces. Our result also provides new spaces satisfying anasymptotic formula and it also builds a link between the asymptotic formula forfunctions and the asymptotic volume ratio of a metric measure space. Inaddition, we prove two related rigidity results for metric measure spaces withsynthetic Ricci curvature bound which are new even in the smooth setting.

On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: asymptotic volume ratio, volume entropy and rigidity