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 - ∪ r ∈ R OK r is the star of r in K. Let Sₐ ,a ∈ M, be a finite set and SA = ∏ a ∈ A Sₐ ,A ⊂ M. A measure μ 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, μ M (xR ) ≠ 0 implies \[ μ _M (X_A × X_B |x_R ) ≠ μ _M (X_A |x_R )μ _M (X_B |x_R ) \]for arbitrary XA ⊂ SA and XB ⊂ SB. Theorem. If the complex K is regular, any consistent family of measures μ K = \ μ K ∈ K on SK = \ SK ∈ K has a unique extension which is Markov relative to K.