Two inequalities for conditional expectations and convergence results for filters

In this paper we prove, first of all, two inequalities for conditional expectations, from which we easily deduce a result by Landers and Rogge. Then we prove convergence results for conditional expectations of the form Pn[f(Xn)|Yn] to a conditional expectation of the form P[f(X)|Y]. We study, in particular, the case in which the random variables Yn,Y are of the type hn(Xn),h(X).