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, Hytönen-Pérez
type and Lerner-Moen type norm estimates for Doob maximal operator are
established. Our approaches are mainly based on the construction of principal
sets.
