  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 (R^n,mu) with mu(B(x,r))<Cr^d, 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.
