ABSTRACT
REMARK 2.2. Suppose that a 11 are strictly increasing a.s. Then if the initial distribution 11 is nonatomic (i.e., JL( {x}) = 0 'Vx or, equivalently, the d.f. of 11 is continuous), then 11 o 'Y_, is nonatomic 'V')' E r (outside a set of zero P-probability). It follows that if X0 has a continuous d.f., then so has X1 and, in turn, X2 has a continous d.f, and so on. Since, by Theorem 2.1, this sequence of continuous d. f.'s (of X11 (n :::: I)) converges uniformly to the d.f. of 1r, the latter is continuous. Thus 1r is nonatomic if a 11 are strictly increasing a.s.
