Weak A_\infty weights and weak Reverse H\"older property in a space of homogeneous type

In the Euclidean setting, the Fujii-Wilson-type $A_\infty$ weights satisfy a Reverse H\"older Inequality (RHI) but in spaces of homogeneous type the best known result has been that $A_\infty$ weights satisfy only a weak Reverse H\"older Inequality. In this paper, we compliment the results of Hyt\"onen, P\'erez and Rela and show that there exist both $A_\infty$ weights that do not satisfy an RHI and a genuinely weaker weight class that still satisfies a weak RHI. We also show that all the weights that satisfy a weak RHI have a self-improving property but the self-improving property of the strong Reverse H\"older weights fails in a general space of homogeneous type. We prove most of these purely non-dyadic results using convenient dyadic systems and techniques.