In the paper Existence de processus Markoviens pour des systèmes infinis de particules by Cocozza, C. and Kipnis, C. (Ann. lnst. H. Poincaré, Sect. B, 13, 239-257, 1977), one reads
A transition probability from $H_1$ to $H_2$ is a function $P: H_1 \times A_2 \to [0,1] $ such that
The map $h_1 \mapsto P(h_1,C)$ is $A_1$ measurable for every $C \in A_2$
the map $C \mapsto P(h_1,C)$ is a probability measure for every $h_1 \in H_1$
Consider the function $X_{h_1}:h_2 \mapsto X(h_1, h_2)$. It is assumed that $$\Bbb{E} [X_{h_1} \mid B_2] = Y_{h_1} $$ $\mu$ - almost surely, where $\mu$ is a probability measure on $H_1$
It seems that if we define $\Bbb{P}(dh_1,dh_2) = \mu(h_1) P(h_1, dh_2)$ we obtain
$$ \Bbb{E}_{\Bbb{P}}[X] = \int_{H_1} \int_{H_2}X_{h_1}(h_2)P(h_1,dh_2) d\mu(h_1)= \int_{H_1} \int_{H_2}Y_{h_1}(h_2)P(h_1,dh_2) d\mu(h_1)= \Bbb{E}_{\Bbb{P}}[Y] $$
But how do we obtain
$$\Bbb{E}_{\Bbb{P}}[X] = Y ?$$
Edit: Perhaps the question is a notation question: what is the meaning of $$\Bbb{E}^{B_2\otimes A_1}_{\Bbb{P}}[X]?$$
Translation: Let $(H_1,A_1)$ and $(H_2,A_2)$ be two measurable spaces and let $B_2$ be a subsigma-algebra of $A_2$. Let $P(h_1,dh2)$ be a transition probability from $H_1$ to $H_2$, and let $\mu$ be a probability on $H_1$. If for $\mu$ - almost every $h_1$ the conditional expectation of a random variable $X(h_1,\cdot)$ with respect to the subsigma algebra $B_2$ and the probability $P(h_1,\cdot)$ is $Y(h_1,\cdot)$ then $$\Bbb{E}^{B_2\otimes A_1}_{\Bbb{P}}[X]$$
where $\Bbb{P}(dh_1,dh_2) = \mu(dh_1) P(h_1, dh_2)$