# Why are discrete random measures not dominated?

A statement I’ve seen without proof in many books and papers is that random probability measures obtained by normalizing a completely random measure, such as the Dirichlet process, are not dominated by a common $$\sigma$$-finite measure and therefore the posterior can’t be obtained using Bayes theorem. I can intuitively grasp why is that but don’t know how to prove it.

Suppose you have a Normalized Random Measure of the form $$P(\cdot) = \sum_h w_h \delta_{\phi_h}(\cdot)$$ with the usual condition that the weights sum to $$1$$ and that $$\phi_h$$ are i.i.d from a non-atomic distribution (non atomicity is important)
Now suppose that you observe a single datum $$X_1 = x^*$$, living in a Polish space $$(\mathbb X, \mathcal X)$$. $$X_1 \mid P \sim P \quad P \sim \Pi$$ with $$\Pi$$ a measure on $$\mathbb P_X$$ the space of all probability measures on $$(\mathbb X, \mathcal X)$$.
Since the $$\phi_h$$s come from a diffuse measure, this entails that the set $$A = \{\text{random measures that have an atom in x^*}\} \subset \mathbb P_X$$ will have $$0$$ prior mass, i.e. $$\Pi(A) = 0$$.
On the other hand, we know that if $$\Pi$$ is for example a Dirichlet process, $$\Pi(A \mid X_1 = x^*) = 1$$.