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