Sparse domination on non-homogeneous spaces with an application to $A_p$ weights

We extend Lerner's recent approach to sparse domination of Calder\'on--Zygmund operators to upper doubling (but not necessarily doubling), geometrically doubling metric measure spaces. Our domination theorem is different from the one obtained recently by Conde-Alonso and Parcet and yields a weighted estimate with the sharp power $\max(1,1/(p-1))$ of the $A_p$ characteristic of the weight.