Weighted estimates for positive operators and Doob maximal operators on filtered measure spaces

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.