This paper contains a study of attractors in cellular automata, particularly the minimal attractors as defined by J Milnor Milnor's definition of attractor uses a measure on the state space, the measures that we consider are Bernoulli product measures Given a Bernoulli measure it is shown that a cellular automaton has at most one minimal attractor, when there is one, it is the omega-hmit set of almost all points Examples are given to show that the minimal attractor can change as the Bernoulli measure is varied Other examples illustrate the difference between this result and the corresponding result that is obtained by replacing Milnor's definition of attractor by the purely topological definition used by C Conley The examples also show that certain invariant sets of cellular automata are less well-behaved than one might hope for instance the periodic points are not necessarily dense in the nonwandenng set Over the last few years there has been a great deal of interest among applied scientists concerning cellular automata One of the reasons for their interest is that numerical studies give evidence that many cellular automata exhibit 'self-organizing behavior' The meaning of this is that for certain automata, a sequence of iterates often appears to have a limiting state that is independent of the choice of initial condition [11] This is the second in a series of papers that are aimed at describing this self-organization In the first paper [8] the ergodicity of the underlying Bernoulli shift is exploited to partially explain the phenomenon of self-organization The explanation in [8] is given in the terminology of topological dynamics, using C Conley's concepts of chain recurrence and attractor We may loosely describe one of the results as follows, precise statements and definitions are given below THEOREM A ([8]) Suppose that ^ is a Bernoulli probability measure A cellular automaton has at most one minimal topologically attracting set with respect to /x, if there is such a set then it contains the omega limit set of x for /x-almost all x The motivation for the current paper is to make better use of the measure theoretic properties of the shift in order to refine Theorem A In particular we replace Conley's topological notion of an attracting set with a measure theoretic formulation due to t Research supported in part by NSF grant DMS-8800758