Commutativity properties of conditional distributions and Palm measures

Given a probability space (Ω,A, P ) and some σ-fields F ,G ⊂ A, we consider iterated conditional distributions of the form (P [ · | F ])[ · | G]. Thus, in the second step, we form the conditional distribution with respect to G, using P [ · |F ] instead of P as the underlying probability measure. Under suitable regularity conditions, we show that conditioning with respect to F and G commute. The commutativity property remains valid for the operations of conditioning on a σ-field F and of forming the Palm distributions with respect to a random measure ξ. This enables us to construct the Palm measures of ξ via conditioning on a suitable σ-field.