Let \((\mathcal{X}, d)\) be a Polish space. Let \(\mu : \Omega \to \mathcal{P}(\mathcal{X})\) be measureable wiht respect to the cylinder \(\sigma \)-algebra on \(\mathcal{P}(\mathcal{X})\). Let \(f : \mathcal{X} \to \mathcal{Y}\) be measurable. define \(\varphi : \Omega \to \mathcal{Y}\) by

Then \(\varphi \) is a measurable map from \(\Omega \to \mathcal{Y}\).