My understanding of the definition of a statistical model is a probability space $(\Omega,\Sigma,p)$, a random variable $X:\Omega\to\mathbb{R}$, and a collection of probability measures $\mu_{\theta}$ on $\mathbb{R}$, where $\theta$ is a parameter.
Aren't the $\mu_{\theta}$ supposed to be related to the random variable though? Are these $\mu_{\theta}$ supposed to be interpreted as different push-forward measures of $p$ by $X$, which would imply, that the probability measure on the original space is parameterized too $p_{\theta}$?
