A conditional product measure theorem
The probabilistic concept of independence is strongly linked to product measures. This paper generalizes this relation to conditional independence. Let \((\Omega_i,{\mathcal S}_i)\) be measurable spaces, \(i=0,1,2\), and let \(\mu_{01}\) and \(\mu_{02}\) be probability measures on \({\mathcal S}_0\otimes{\mathcal S}_1\) and \({\mathcal S}_0\otimes{\mathcal S}_2\), respectively, which give on \({\mathcal S}_0\) the same marginal measure. Let \(P^i(A_i)\), \(i=1,2\), be the conditional probabilities of \(A_i\in{\mathcal S}_i\) given \(\omega_0\in\Omega_0\). Whenever at least one of these two conditional probabilities has a regular version, then there exists a unique probability measure \(\mu\) on \({\mathcal S}_0\otimes{\mathcal S}_1\otimes{\mathcal S}_2\) which has the marginal measures \(\mu_{01}\) and \(\mu_{02}\) and such that for all \(A_1\in{\mathcal S}_1\), \(A_2\in{\mathcal S}_2\) the conditional probability \(P^{12}(A_1\times A_2)\) is equal to the product \(P^1(A_1)P^2(A_2)\), i.e., the variables \(\omega_1\) and \(\omega_2\) are conditionally independent. It is shown by an example that this result does not hold when neither of the \(P^i(A)\) has a regular version.
scientific article; zbMATH DE number 896121