I am trying to set up a probabilistic framework that can be easily switched between a frequentist and a Bayesian approach.
Assume the measurable spaces $(\mathcal{X}, \mathfrak{X}), (\mathcal{T}, \mathfrak{T})$ and $(\Omega, \mathfrak{F})$, where $\Omega = \mathcal{X} \times \mathcal{T}$ denotes the sample space equipped with the $\sigma$-algebra $\mathfrak{F}=\mathfrak{X} \otimes \mathfrak{T}$. Let $X: \Omega \to \mathcal{X}$ and $T: \Omega \to \mathcal{T}$ be measurable functions, where $T$ represents a parameter.
I want to assume that the distribution of $X$ is parametrized by $T=t$. So to formalize this distribution, I started by defining a probability kernel $P: \mathcal{T} \times \mathfrak{X} \to [0,1]$, such that $P(t, A) = P_t(A)$ is $\mathfrak{T}$-measurable in $t \in T$ for fixed $A \in \mathfrak{X}$ and a probability measure in $A \in \mathfrak{X}$ for fixed $t \in T$. This leads to the probability space $(\mathcal{X}, \mathfrak{X}, (P_t)_{t \in \mathcal{T}})$.
In the Bayesian setting, I would have defined a prior measure $\mu_T$ on $(\mathcal{T}, \mathfrak{T})$ and used it to extend $P$ to a regular conditional distribution as well as define a probability measure $\mathbb{P}$ on $(\Omega, \mathfrak{F})$ by \begin{align*} \mathbb{P}[X \in A, T \in B] = \int\limits_B P(t, A) \mu_T(dt), \quad A \in \mathfrak{X}, B \in \mathfrak{T}. \end{align*}
Q1: Would this be reasonable? If not, what would be an elegant way to "measure-theoratically" introduce the distribution of $X$ under a parametrization $T=t$ that allows for flexibility on the parameter structure?
Q2: What happens if $T$ is non-random? Can/should I define a regular conditional distribution in the frequentist case (maybe with the Dirac measure on $(\mathcal{T}, \mathfrak{T})$)?