Markov Measures and Markov Extensions

Let K be a complex with the set of vertices M and A, B and R three subsets of M. R is said to be separating A and B in K (notation: (A |R B) K if any a A and b B are not connected in K - ₑ ₑ OK r is the star of r in K. Let Sₐ, a M, be a finite set and SA = ₀ ₀ Sₐ, A M. A measure M on SM is said to be Markov relative to K if for any separation (A |R B) K if any a A and a A and xR SR the inequality, M (xR) 0 implies \ M (XA XB |xR) M (XA |xR) M (XB |xR) arbitrary XA SA and XB SB. Theorem. If the complex K is regular, any consistent family of measures K = \ { K \}₊ ₊ on SK = \ {SK \}₊ ₊ has a unique extension which is Markov relative to K.