The John–Nirenberg type inequality for non-doubling measures
X. Tolsa defined a space of BMO type for positive Radon measures satisfying some growth condition on $\mathbb R^d$. This new BMO space is very suitable for the Calderón–Zygmund theory with non-doubling measures. Especially, the John–Nirenberg type inequality can be recovered. In the present paper we introduce a localized and weighted version of this inequality and, as applications, we obtain some vector-valued inequalities and weighted inequalities for Morrey spaces.