  We characterize strong type and weak type inequalities with two weights for
positive operators on filtered measure spaces. These estimates are
probabilistic analogues of two-weight inequalities for positive operators
associated to the dyadic cubes in $\mathbb R^n$ due to Lacey, Sawyer and
Uriarte-Tuero \cite{LaSaUr}. Several mixed bounds for the Doob maximal operator
on filtered measure spaces are also obtained. In fact, Hyt\"{o}nen-P\'{e}rez
type and Lerner-Moen type norm estimates for Doob maximal operator are
established. Our approaches are mainly based on the construction of principal
sets.
