Let $ (\mathbb{X} _i, \mathscr{X}_i) $ and $ (\mathbb {Y} _i, \mathscr {Y} _i) $ measurable spaces with $ i = 1, 2 $. Let $ \gamma_i: \mathscr {X}_i\times\mathbb {Y}_i\longrightarrow [0,1] $ a probability kernel from $ (\mathbb {Y}_i, \mathscr {Y}_i) $ to $ (\mathbb {X}_i , \mathscr {X}_i) $, i.e., $ \gamma_i(X_i|\;\;\;\;):\mathbb {Y}_i\longrightarrow [0,1] $ is a $\mathscr{Y}_i$-measurable function for all $X_i\in\mathscr{X}_i$ and $ \gamma_i(\;\;\;\;|\omega_i): \mathscr {X}_i\longrightarrow [0,1] $ is a probability for all $\omega_i\in\mathbb{Y}_i$.
Let $ \mathbb {X} = \mathbb {X} ^ 1 \times \mathbb{X}^ 2 $, $ \mathbb {Y} = \mathbb{Y}^1\times\mathbb{Y}^2$. Denote by $\mathscr{X}=\sigma(\mathscr{X}^1\times\mathscr{X}^2)$ a $\sigma$-algebra generated by the algebra $\mathscr{X}^1\times\mathscr{X}^2$ and $\mathscr{Y}=\sigma(\mathscr{Y}^1\times\mathscr{Y}^2)$ a $\sigma$-algebra generated by the algebra $\mathscr{Y^1}\times\mathscr{Y}^2$. Set $\gamma_1 \times \gamma_2: \mathscr {X}\times\mathbb {Y} \longrightarrow [0,1] $ for $ X \in \mathscr {X} $ and $ \omega = (\omega_1, \omega_2) \in \mathbb{Y}_1 \times \mathbb{Y}_2 = \mathbb{Y} $ making $ \gamma_1\times\gamma_2(\;\cdot\;|\omega)=\gamma_1(\;\cdot\;|\omega_1)\times\gamma_2(\;\cdot\;|\omega_2) $ and $ \gamma_1\times\gamma_2(X|\omega)=\gamma_1(\;\cdot\;|\omega_1)\times\gamma_2(\;\cdot\;|\omega_2)(X) $ It is interesting to raise the following question. Under what conditions can we say that $ \gamma = \gamma_1\times\gamma_2$ is a probability kernel from $ (\mathbb {Y}, \mathscr {Y}) $ to $ (\mathbb {X} , \mathscr {X}) $?