Weak Gurov-Reshetnyak class in metric measure spaces

We introduce a weak Gurov-Reshetnyak class and discuss its connections to a weak Muckenhoupt $A_\infty$ condition and a weak reverse H\"older inequality in the setting of metric measure spaces with a doubling measure. A John-Nirenberg type lemma is shown for the weak Gurov-Reshetnyak class which gives a specific decay estimate for the oscillation of a function. It implies that a function in the weak Gurov-Reshetnyak class satisfies the weak reverse H\"older inequality. This comes with an upper bound for the reverse H\"older exponent depending on the Gurov-Reshetnyak parameter which allows the study of the asymptotic behavior of the exponent.