0. We consider transition probability operators P(x,E) on a compact metric space which satisfy a rather strong condition (uniform stability in mean). ?1 is devoted largely to notation and terminology. In ?2 an ergodic decomposition is obtained. It is essentially a sharpening of a decomposition of Yosida's valid under less stringent conditions [17]. In ?3 we discuss the behavior of sample paths of the Markov processes themselves relative to the ergodic decomposition obtained in ?2. In ?4 we give some examples of transition probability operators which are uniformly stable in mean. Most of the results of ?2 and some of the results of ?4 appeared in the author's thesis, written under the guidance of Professor Blackwell. Tbe results of ?3 had to await Breiman's discovery of a new version of the strong law of largenumbers. Our Theorem 3.2 is a minor generalization of Breiman's theorem and our proof a slight modification of his. 1. S is a compact metric space. Its metric is denoted by d. , is the a-field generated by the open subsets of S. C(S) is the Banach space of real-valued continuous functions on S with the uniform norm. The dual C*(S) consists of all the finite countably additive measures on S. The value of a member p of C*(S) at a member f of C(S) will be denoted by either (f, ,u) or f f dy. A countably additive measure M on any a-field over any set Q is called a probability measure if it is non-negative and if ,u(Q) = 1. We sometimes refer to probability measures in C*(S) as distributions. A function P on S x I is called a transition probability operator if (i) P(x, * ) is a probability distribution for each x E S, (ii) P( , E) is a s-measurable function for each E e Z. Unless otherwise stated, we will from now on assume that P is a fixed transition probability operator. We will abuse the language by using 'P(x,E)' rather than 'P' to denote not only a value of P but P itself. Associated with P(x,E) are operators T and U, the first defined on bounded X-measurable functions f by (Tf)(x)