On the independence of correspondences (Q2701625)

fet \((T,L({\mathcal T}),L(\lambda))\) and \((\Omega ,L({\mathcal A}),L(P))\) be Loeb probability spaces, and \(X\) a Polish space. By a set-valued process we mean a closed-valued measurable correspondence \(F\) from the Loeb product space \((T\times\Omega ,L({\mathcal T}\otimes {\mathcal A}),L(\lambda\otimes P))\) into \(X\). Such a process is almost independent if for \(L(\lambda\otimes\lambda)\)-almost all \((s,t)\in T\times T\) the random sets \(F(s,\cdot)\) and \(F(t,\cdot)\) are independent. The main result of the paper says that an almost independent set-valued process \(F\) has a Castaing representation consisting of almost independent processes, i.e., there exists a sequence \((f^{i})\) of measurable selections of \(F\) such that for almost all \((t,\omega)\in T\times\Omega\), \(F(t,\omega)=\text{cl}\{f^{i}(t,\omega) : i=1,2,\dots\}\), and for any fixed \(i\) the process \(f^{i}\) is almost independent. Then the author proves the equivalence of different definitions of almost independence for set-valued processes. The proofs use some recent results of \textit{Y. N. Sun} [``The almost equivalence of pairwise and mutual independence and the duality with exchangeability'', Probab. Theory Relat. Fields 112, No. 3, 425-456 (1998; Zbl 0917.60041); ``The complete removal of individual uncertainty: multiple optimal choices and random exchange economies'', Econom. Theory 14, No. 3, 507-544 (1999; Zbl 0957.91064)].