Markov random fields

Let A1, ℬ, A2 be σ-algebras of events having the following relationship: if the outcomes of all events in ℬ are known, events A2 ∈A2 are independent of events A1 ∈ A1. More precisely, the σ-algebras A1 and A2 are conditionally independent with respect to ℬ; this gives the equation for conditional probabilities: $$ P({A_1} \cdot {A_2}|B) = P({A_1}|B) \cdot P({A_2}|B) $$ (1.1) for any A1 ∈ A1, A2 ∈A2. We say that the σ-algebra ℬ splits A1 and A2 (or is splitting) if (1.1) holds for A1, ℬ, A2.