Here is how I understand the standard parametric model in statistical inference:
We have a r.v. $X:\Omega \to \Psi$ which has some known to us distribution yet the exact parameter is unknown to us. So we know that $\text{Law}(X)\in(\mu_\vartheta)_{\vartheta \in \Theta}$. How we have $X_1, \ldots, X_n$ which are i.i.d. copies of $X$. We build some estimator $T_n: \Psi^n\to\Theta$ which should allow us to estimate the real $\vartheta \in \Theta$ by looking at some sample values $(x_1,\ldots, x_n) \in \Psi^n$.
Then we define $\mathbb P_\vartheta := \mu_\vartheta^{\otimes n} $ and all sorts of properties $T_n$ can have. E.g. we call $T_n$ consistent if $$ \forall \vartheta \in \Theta: T_n \xrightarrow{\mathbb P_\vartheta} \vartheta.$$ But how does it make sense if each $T_n$ is defined on a different space and $n$ is also in the definition of $\mathbb P_\vartheta$?
My interpretation doesn't seem to sum up. Any clarifications are hugely appreciated.