I know what a measurable function is but sometimes I see $\Sigma$-measurable function where $\Sigma$ is some sigma algebra which I don't understand because you need two sigma algebras to define a measurable function.
For example: from Wikipedia
Let $(X,\mathcal A)$ and $(Y,\mathcal B)$ be measurable spaces. A Markov kernel with source $(X,\mathcal A)$ and target $(Y,\mathcal B)$ is a map $\kappa : \mathcal B \times X \to [0,1]$ with the following properties:
For every (fixed) $B \in \mathcal B$, the map $ x \mapsto \kappa(B,x)$ is $\mathcal A$-measurable
For every (fixed) $ x \in X$, the map $ B \mapsto \kappa(B, x)$ is a probability measure on $(Y, \mathcal B)$
Now, when it says that $ x \mapsto \kappa(B,x)$ is $\mathcal{A}$-measurable, what is the second sigma algebra? is it $\mathcal{B}$?
$x\mapsto \kappa(B,x),\;\;\mathcal{A}$-measurable means that $\kappa$ is a function of $x\in X$. Since it maps to $[0,1]$ then the second $\sigma-$algebra is $\text{Borel}([0,1])$ so we have:
$$k(x|B):(X,\mathcal{A}) \to ([0,1],\text{Borel}([0,1]))$$