  Carleson measures are ubiquitous in Harmonic Analysis. In the paper of
Fefferman--Kenig--Pipher in 1991 an interesting class of Carleson measures was
introduced for the need of regularity problems of elliptic PDE. These Carleson
measures were associated with $A_\infty$ weights. In discrete setting (we need
exactly discrete setting here) they were studied by Buckley's, where they were
associated with dyadic $A\infty^d$. Our goal here is to show that such
Carleson--Buckley measures (in discrete setting) exists for virtually any
positive function (weight). Of course some modification is needed, because it
is known that Carleson property of Buckley's measure are equivalent to the
weight to be in $A_\infty^d$. However a very natural generalization of those
facts exist for weights more general $A_\infty$, and of course, in a special
case of $A_\infty^d$ it gives Buckley's results. Our generalization of
Buckley's inequality beyond the scope of $A_\infty$ allows us to prove the
so-called bump conjecture.
