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A new class of metric measure spaces is introduced and studied. This class generalises the well-established doubling metric measure spaces as well as the spaces (Rn, μ) with μ(B(α, r)) ≤ Crd, in which non-doubling harmonic analysis has recently been developed. It seems to be a promising framework for an abstract extension of this theory. Tolsa’s space of regularised BMO functions is defined in this new setting, and the John-Nirenberg inequality is proven

A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa