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 R n due to Lacey, Sawyer and Uriarte-Tuero [30]. 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.

For a general dyadic grid, we give a Calderón–Zygmund type decomposition, which is the principle fact about the multilinear maximal function frakturM on the upper half‐spaces. Using the decomposition, we study the boundedness of frakturM. We obtain a natural extension to the multilinear setting of Muckenhoupt's weak‐type characterization. We also partially obtain characterizations of Muckenhoupt's strong‐type inequalities with one weight. Assuming the reverse Hölder's condition, we get a multilinear analogue of Sawyer's two weight theorem. Moreover, we also get Hytönen–Pérez type weighted estimates.

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