Non-homogeneous T b theorem and random dyadic cubes on metric measure spaces

We prove a T b theorem on quasimetric spaces equipped with what we call an upper doubling measure. This is a property that encompasses both the doubling measures and those satisfying the upper power bound μ(B(x, r)) ≤ Cr ^d . Our spaces are only assumed to satisfy the geometric doubling property: every ball of radius r can be covered by at most N balls of radius r/2. A key ingredient is the construction of random systems of dyadic cubes in such spaces.