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...

Let G be a group acting on Ω and ℱ a G-invariant algebra of subsets of Ω. A full conditional probability on ℱ is a function P: ℱ × (ℱ∖{∅}) → [0,1] satisfying the obvious axioms (with only finite additivity). It is weakly G-invariant provided that P(gA|gB) = P(A|B) for all g ∈ G and A,B ∈ ℱ, and strongly G-invariant provided that P(gA|B) = P(A|B) whenever g ∈ G and A ∪ gA ⊆ B. Armstrong (1989) claimed that weak and strong invariance are equivalent, but we shall show that this is false and that weak...