I'm writing a text on Allan Birnbaum's proof that the likelihood principle is implied by the conjunction of the sufficiency principle and the conditionality principle.
Here, we call
\begin{equation}
E=\left( \mathcal{X}, \Sigma_{\mathcal{X}}, \lbrace p_\theta:\mathcal{X} \to \mathbb{R} \middle| \theta \in \Theta \rbrace; \mu \right)
\end{equation}
an experiment or model, where $\mathcal{X}$ is a sample space with possible outcomes, $\Sigma_{\mathcal{X}}$ a $\sigma$-algebra, and $\lbrace p_\theta:\mathcal{X} \to \mathbb{R} | \theta \in \Theta \rbrace$ a collection of probability density functions with respect to the support measure $\mu:\Sigma_{\mathcal{X}} \to [0, \infty]$, that induce the collection of probability measures $\{P_\theta: \Sigma_{\mathcal{X}}\to[0,1]|\theta \in \Theta\}$. A statistic $T: \mathcal{X} \to \mathcal{Y}$ is a measurable function. A statistic is sufficient for $\Theta$ or $E$ when, for all $\theta \in \Theta$ the conditional expectation $\mathbb{E}_{\theta}(1_A|T)$ does not depend on $\theta$. When Birnbaum introduces the principle of sufficiency, he introduces the concept of an experiment $E'$ that is derived by $E$ and a corresponding sufficient statistic $T$. That is, whenever the outcome $x \in \mathcal{X}$ is 'observed' in $E$, the corresponding outcome $T(x) \in T(\mathcal{X}) \subset\mathcal{Y}$ is observed in $E'$. Now my question is, can $E'$, which is not well-defined, be an actual experiment or model?
When I started thinking about this, I concluded that the sample space of $E'$ must be $T(\mathcal{X})$ and its $\sigma$-algebra the set $\Sigma_{T(\mathcal{X})} := \{A \cap T(\mathcal{X})|A \in \Sigma_{\mathcal{Y}}\}$. To have the same probabilistic structure as $E$, we must consider the pushforward measures $P_\theta T^{-1}$ and $\mu T^{-1}$, defined by $\mu T^{-1}(A)= \mu(T^{-1}(A)),~A\in \Sigma_{T(\mathcal{X})}$, and $P_\theta T^{-1}$ analogously. Now, my more specified question is, are there probability density functions $p_\theta^T:T(\mathcal{X}) \to \mathbb{R}$, such that $p_\theta^T$ induces $P_\theta T^{-1}$ with respect to $\mu T^{-1}$? If this is the case, $E'$, with this explanation, can be a well-defined mathematical object.
Thanks in advance!