A discrete stochastic Korovkin theorem (Q1187064)

Given a probability space \((\Omega,{\mathcal F},P)\) and a countable set \(T\), denote by \(B_ 0(T)\) the vector space of all stochastic processes \(X(T,\omega)\) with real state space and index set \(T\) satisfying \[ \sup_{t\in T}\int| X(t,\omega)| dP(\omega)<+\infty. \] The space \(B_ 0(T)\) is endowed with the pointwise convergence in the first mean. Suitable conditions of Korovkin type are stated in order that an arbitrary sequence \((L_ n)_{n\in N}\) of positive linear operators from \(B_ 0(T)\) to \(B_ 0(T)\) converge to the identity operator, i.e., \(L_ n(X)\to X\) in \(B_ 0(T)\), for every \(X\in B_ 0(T)\).