Let \(X\), \(Y\), \(Z\) be random variables on some probability space. Recall that
\[ P(X \in A \mid Y, Z) = P(X \in A \mid \sigma \{Y, Z\}) \]
is the conditional expectation of the indicator function of the event \(\{X \in A\}\), given the \(\sigma \)-algebra \(\sigma \{Y, Z\}\) generated by \(Y\) and \(Z\). We say that \(X\) and \(Y\) are independent, given \(Z\), if for all Borel sets \(A, B \subset \mathbb {R}\),
\[ P(X \in A, Y \in B \mid Z) = P(X \in A \mid Z) \cdot P(Y \in B \mid Z) \quad P\text {-almost surely}. \]
Show that this condition is equivalent to having \(P(X \in A \mid Y, Z) = P(X \in A \mid Z)\) \(P\)-almost surely, for all Borel sets \(A\).