  In a filtered measure space, a characterization of weights for which the
trace inequality of a positive operator holds is given by the use of discrete
Wolff's potential. A refinement of the Carleson embedding theorem is also
introduced. Sawyer type characterization of weights for which a two-weight norm
inequality for a generalized Doob's maximal operator holds is established by an
application of our Carleson embedding theorem. Moreover, Hyt\"{o}nen-P\'{e}rez
type one-weight norm estimate for Doob's maximal operator is obtained by the
use of our two-weight characterization.
