Let $\Theta$ and $X$ be two (Hausdorff) topological spaces. Let $\mathbb P : \Theta \to \Delta(X)$ be a "statistical model", i.e., a continuous function from parameter space $\Theta$ to the space of probability measures $\Delta(X)$ (equipped with the topology of weak convergence of measures).

Let $S_\theta \subseteq X$ denote the support of the measure $\mathbb P_\theta$ (i.e., the smallest closed set of full measure). Define the relation $$\Theta' := \{ (\theta, x) : x \in S_\theta \}.$$ Is $\Theta'$ a closed subset of the product space $\Theta \times X$? If not, is $\Theta'$ at least measurable?

**Edit:** fedja points out that $\Theta'$ need not be closed. In that example, $\Theta' \subseteq [0,1] \times \mathbb R$, and $\Theta' = \big\{(0,0)\big\} \cup \big( (0,1] \times \mathbb R \big)$.