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We generalize Bourgain-Brezis-Mironescu's asymptotic formula for fractionalSobolev functions, in the setting of abstract metric measure spaces, under theassumption that at almost every point the tangent space in the measuredGromov-Hausdorff sense is a finite dimensional Banach space or a Carnot group.Our result not only covers the known results concerning Euclidean spaces,weighted Riemannian manifolds and finite dimensional Banach spaces, but alsoextends Bourgain-Brezis-Mironescu's formula to $\rcdkn$ spaces andsub-Riemannian manifolds.

On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: Bourgain-Brezis-Mironescu's theorem revisited