Convergence of monotone functions

Let P be a strongly-diffuse probability on R(m) and (f(n)) a sequence of functions from R(m) to R(m) such that a) or b) are satisfied: a) There exists a non decreasing sequence (A(p)) with P(A(p)) --> 1 such that, for all p, the functions f(n) are monotone on A(p) and the sequence (f(n)1A(p))(n) converges in sigma(L(1)(P), L(infinity)(P)). b) The functions f(n) are cyclically monotone on D with P(D) = 1 and the sequence (f(n)) converges in law. Then (f(n)) converges P-a.e.