Strong $A_\infty$-weights are $A_\infty$-weights on metric spaces

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.