ANNALES POLONICI MATHEMATICI 96.1 (2009) Markov operators and n-copulas by P. Mikusi´ nski and M. D. Taylor (Orlando, FL) Abstract. We extend the definition of Markov operator in the sense of J. R. Brown and of earlier work of the authors to a setting appropriate to the study of n-copulas. Basic properties of this extension are studied. 1. Introduction. In [3], J. R. Brown introduced Markov operators as positive operators T : L ∞ (X ) → L ∞ (X ) satisfying T 1 = 1 and  X T f dν =  X f dν where (X, A,ν ) was a given probability space. One of his main in- terests was the role played by the particular operators T φ f = f ◦ φ in- duced by invertible measure-preserving φ : X → X . Another was in the fact that under the constraints he imposed, there was a one-to-one cor- respondence between the set of Markov operators T on X and the set of doubly stochastic measures μ on X × X . (To say that μ was doubly stochastic in this setting meant that for measurable sets A of X , one had μ(A × X )= μ(X × A)= ν (A).) It would appear that an inspiration for Brown’s definition of Markov op- erator was the theory of Markov processes. One might think of T : L ∞ (X ) → L ∞ (X ) as describing the “evolution”, f → Tf , of a function over a fixed time interval. However, we found Brown’s work interesting because of its applicability to a different mathematical scenario. Brown required his probability measure ν to be nonatomic, which im- plied, by results on Borel equivalences (see, for example, [14]), that one could take X to be the unit interval, I = [0, 1], ν could be taken to be λ, 1-dimensional Lebesgue measure on I , and μ would be a doubly stochastic measure on I 2 .A 2-copula is a function C : I 2 → I which is related to a doubly stochastic measure μ on I 2 by C (x 1 ,x 2 )= μ([0,x 1 ] × [0,x 2 ]). (C may be regarded as the joint distribution function of two random variables that 2000 Mathematics Subject Classification : Primary 47N30; Secondary 60E05, 62H05. Key words and phrases : copula, Markov operator, conditional expectation. DOI: 10.4064/ap96-1-7 [75] c  Instytut Matematyczny PAN, 2009