  In this note we generalize the definition of Fujii-Wilson condition providing
quantitative characterizations of some interesting classes of weights, such as
$A_\infty$, $A_\infty^{weak}$ and $C_p$, in terms of BMO type spaces suited to
them. We will provide as well some self improvement properties for some of
those generalized BMO spaces and some quantitative estimates for Bloom's BMO
type spaces.
