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two-dimensional probabilities; extremal measure
A properly measurable set ${\cal P} \subset {X} \times M_1({Y})$ (where ${X}, {Y}$ are Polish spaces and $M_1(Y)$ is the space of Borel probability measures on $Y$) is considered. Given a probability distribution $\lambda \in M_1(X)$ the paper treats the problem of the existence of ${X}\times {Y}$-valued random vector $(\xi ,\eta )$ for which ${\cal L}(\xi )=\lambda $ and ${\cal L}(\eta | \xi =x) \in {\cal P}_x$ $\lambda $-almost surely that possesses moreover some other properties such as “${\cal L}(\xi ,\eta )$ has the maximal possible support” or “${\cal L}(\eta | \xi =x)$’s are extremal measures in ${\cal P}_x$’s”. The paper continues the research started in [7].
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