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We study the so-called John-Nirenberg space that is a generalization offunctions of bounded mean oscillation in the setting of metric measure spaceswith a doubling measure. Our main results are local and global John-Nirenberginequalities, which give weak type estimates for the oscillation of a function.We consider medians instead of integral averages throughout, and thus functionsare not a priori assumed to be locally integrable. Our arguments are based on aCalder\'{o}n-Zygmund decomposition and a good-$\lambda$ inequality for medians.A John-Nirenberg inequality up to the boundary is proven by using chainingarguments. As a consequence, the integral-type and the median-typeJohn-Nirenberg spaces coincide under a Boman-type chaining assumption.

Median-type John-Nirenberg space in metric measure spaces