Regular conditional Bayesian experiment In "Elements of Bayesian Statistics" (1990), Florens, Mouchart and Rolin describe two basic forms of reduction of a Bayesian experiment: Marginalization and Conditioning (Ch. 1). I don't understand the conditioning reduction. More precisely, i struggle with the definition of a regular conditional experiment. I would appreciate an explanation, if possible in measure-theoretic terms. Thanks
 A: Let us consider a more general case with arbitrary $\sigma$-fields $\mathcal{B}\subseteq\mathcal{A}$, $\mathcal{T}\subseteq\mathcal{S}$ and $\mathcal{M}\subseteq\mathcal{A}\otimes\mathcal{S}$ and a combined conditional/marginal experiment
$$
\mathcal{E}^\mathcal{M}_{\mathcal{B}\vee\mathcal{T}}=\left(A\times S,\mathcal{A}\otimes\mathcal{S},\Pi^\mathcal{M}_{\mathcal{B}\vee\mathcal{T}}\right)
$$
(see "Combined Reductions" on pp. 53-54).
Define
$$
\begin{array}{lcl}
\Omega & := & A\times S \\
\Xi & := & \mathcal{A}\otimes\mathcal{S} \\
\Gamma & := & \mathcal{B}\vee\mathcal{T}\vee\mathcal{M} \\
\Gamma' & := & \left(\Omega,\Gamma\right) \\
\Delta_1 & := & \left(\mathcal{B}\vee\mathcal{M}\right)\otimes\left(\mathcal{T}\vee\mathcal{M}\right) \\
\Delta_1' & := & \left(\Omega\times\Omega,\Delta_1\right) \\
\Delta_2 & := & \left(\mathcal{T}\vee\mathcal{M}\right)\otimes\left(\mathcal{B}\vee\mathcal{M}\right) \\
\Delta_2' & := & \left(\Omega\times\Omega,\Delta_2\right)
\end{array}
$$
and define $X$ to be the function
$$
X:\Omega\rightarrow\left(\Omega\times\Omega\right),\hspace{10mm}X\left(\omega\right):=\left(\omega,\omega\right)
$$
Note that $X$ is a $\Gamma/\Delta_1$-measurable as well as $\Gamma/\Delta_2$-measurable.
$\mathcal{E}^\mathcal{M}_{\mathcal{B}\vee\mathcal{T}}$ is called regular iff $\Pi^\mathcal{M}_{\mathcal{B}\vee\mathcal{T}}$ is regular (the book does not make it clear whether $\Pi^\mathcal{M}_{\mathcal{B}\vee\mathcal{T}}$ should be assumed regular when $\mathcal{E}^\mathcal{M}_{\mathcal{B}\vee\mathcal{T}}$ is not regular) and there exist regular versions of $\mu^{\mathcal{T}\vee\mathcal{M}}_\mathcal{B\vee\mathcal{M}}$ and $P^{\mathcal{B}\vee\mathcal{M}}_{\mathcal{T}\vee\mathcal{M}}$.
If this is the case,
$$
\mu^\mathcal{M}_{\mathcal{B}\vee\mathcal{M}}\otimes P^{\mathcal{B}\vee\mathcal{M}}_{\mathcal{T}\vee\mathcal{M}}
$$
is a Markov kernel from $\mathcal{M}$ to $\Delta_1$ and for all $D\in\Delta_1$,
$$
\Pi^\mathcal{M}_{\mathcal{B}\vee\mathcal{T}\vee\mathcal{M}}\left(\omega, X^{-1}(D)\right)=\left(\mu^\mathcal{M}_{\mathcal{B}\vee\mathcal{M}}\otimes P^{\mathcal{B}\vee\mathcal{M}}_{\mathcal{T}\vee\mathcal{M}}\right)\left(\omega,D\right)\tag{1}
$$
for $\left[\Xi,\Pi_\mathcal{M}\right]$-almost every $\omega\in\Omega$. Write $\Pi^\mathcal{M}_{\mathcal{B}\vee\mathcal{M},\mathcal{T}\vee\mathcal{M}}\left(\omega,D\right)$ as a shorthand for $\Pi^\mathcal{M}_{\mathcal{B}\vee\mathcal{T}\vee\mathcal{M}}\left(\omega, X^{-1}(D)\right)$, $D\in\Delta_1$.
In the same vein,
$$
P^\mathcal{M}_{\mathcal{T}\vee\mathcal{M}}\otimes \mu^{\mathcal{T}\vee\mathcal{M}}_{\mathcal{B}\vee\mathcal{M}}
$$
is a Markov kernel from $\mathcal{M}$ to $\Delta_2$ and for all $D\in\Delta_2$,
$$
\Pi^\mathcal{M}_{\mathcal{B}\vee\mathcal{T}\vee\mathcal{M}}\left(\omega,X^{-1}(D)\right)=\left(P^\mathcal{M}_{\mathcal{T}\vee\mathcal{M}}\otimes\mu^{\mathcal{T}\vee\mathcal{M}}_{\mathcal{B}\vee\mathcal{M}}\right)\left(\omega,D\right)\tag{2}
$$
for $\left[\Xi,\Pi_\mathcal{M}\right]$-almost every $\omega\in\Omega$. Write $\Pi^\mathcal{M}_{\mathcal{T}\vee\mathcal{M},\mathcal{B}\vee\mathcal{M}}\left(\omega,D\right)$ as a shorthand for $\Pi^\mathcal{M}_{\mathcal{B}\vee\mathcal{T}\vee\mathcal{M}}\left(\omega, X^{-1}(D)\right)$, $D\in\Delta_2$.
Equations $(1)$ and $(2)$ together constitute the rigorous interpretation of equations $1.4.16$ in Florens et al.'s book. See here for more on that.
$X$ is an embedding of $\Gamma'$ in $\Delta'_1$ in the sense that $X$ is injective and measurable and $\Gamma=X^{-1}\left(\Delta_1\right)$.
Moreover, from Doob's Functional Representation Lemma (Lemma 1.13 in Kallenberg's Foundations of Modern Probability (2002)), also known as The Factorisation Lemma, if $\left(H,\mathcal{B}\right)$ is a Borel space, then $f:\Omega\rightarrow H$ is $\Gamma/\mathcal{B}$-measurable iff there is some $\Delta_1/\mathcal{B}$-measurable function $h$ such that $f=h\circ X$.
Additionally, if we equip $\Gamma'$ with a probability measure $\phi$ (e.g. $\phi=\Pi$ or $\phi=\Pi^\mathcal{M}_{\mathcal{B}\vee\mathcal{T}}\left(\omega,\cdot\right)$ for some fixed $\omega\in\Omega$) and equip $\Delta'_1$ with $X$'s distribution function, $\phi_X$, then $f$ and $h$ have the same distributions. Furthermore, if $\mathcal{B}$ is the standard Borel field on the extended real line, we have by the Substitution Lemma (Lemma 1.22 in Kallenberg's book)$^*$, that $f$ has an integral w.r.t. $\left(\Gamma',\phi\right)$ iff $h$ has an integral w.r.t. $\left(\Delta'_1,\phi_X\right)$ and in this case the values of the integrals are identical.
In like manner, $X$ is an embedding of $\Gamma'$ in $\Delta'_2$.

$(*)$ Actually, Theorem 1.6.12 in Ash's Probability and Measure Theory (2000) is more appropriate for the level of generality stated here.
