We prove that every strong $A_\infty$-weight is a Muckenhoupt weight in Ahlfors-regular metric measure spaces that support a Poincaré inequality. We also explore the relations between various definitions for $A_\infty$-weights in this setting, since some of these characterizations are needed in the proof of the main result.
"Strong $A_\infty$-weights are $A_\infty$-weights on metric spaces." Rev. Mat. Iberoamericana 27 (1) 335 - 354, January, 2011.