In probability theory we had the following proposition about stochastic kernels:
Let $P_1$ be a probability measure on $(\Omega_1, \mathcal{A}_1)$, and K a stochastic kernel from $(\Omega_1, \mathcal{A}_1)$ to $(\Omega_2, \mathcal{A}_2)$. Then, there exists a unique probability measure P on $(\Omega_1 \times \Omega_2, \mathcal{A}_1 \otimes \mathcal{A}_2)$ such that $$P[A_1 \times A_2] = \int_{A_1}K(w_1, A_2)P_1(dw_1)$$ for all $A_1 \in \mathcal{A_1}$, $A_2 \in \mathcal{A_2}$.
My question is about the difference between the direct product for the spaces $\Omega_i$ but the tensor product for the $\sigma-$algebras. When do I have to use which one or what exactly is the difference between them?
The direct product $A\times B$ of two sets is the set of ordered pairs $(a,b)$ where $a\in A$ and $b\in B$. If $\Omega_1$ is a universe (probability space) and $\Omega_2$ is another universe, then $\Omega_1\times \Omega_2$ describes those universes existing in parallel; to specify a point in $\Omega_1\times \Omega_2$, you need to specify the point in both $\Omega_1$ and $\Omega_2$. For example, if $\Omega$ is the probability space which describes the outcome of the roll of a die, then $\Omega\times \Omega$ describes the outcome of two die rolls.
When $(\Omega_1,\mathcal F_1)$ and $(\Omega_2,\mathcal F_2)$ are measruable spaces, then the natural $\sigma$-algebra associated to $\Omega_1\times \Omega_2$ is $\mathcal F_1\otimes \mathcal F_2$, which is defined as the $\sigma$-algebra generated by sets of the form $E_1\times E_2$, with $E_i\in \mathcal F_i$. Roughly, if $\mathcal F_i$ represents the information about $\Omega_i$, then $\mathcal F_1\otimes \mathcal F_2$ is the combined information from these two $\sigma$-algebras about $\Omega_1\times \Omega_2$.
Whereas $\times$ is defined for any two sets, $\otimes$ is only defined for $\sigma$-algebras (in this context), so there is no ambiguity about where to use each of them.