Setting
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $(S, \mathcal{S})$ and $\left(S^{\prime}, \mathcal{S}^{\prime}\right)$ be two measurable spaces. Consider two independent r.v. $X: \Omega \rightarrow S$ and $Y: \Omega \rightarrow S^{\prime}$. a) Let $f: S \times S^{\prime} \rightarrow[0, \infty)$ be a measurable with respect to the product $\sigma$ -algebra $\mathcal{S} \otimes \mathcal{S}^{\prime} .$
Define $$ g: x \in S \mapsto \mathbb{E}[f(x, Y)] \in[0, \infty] $$ I want to show that $\mathbb{E}[g(X)]=\mathbb{E}[f(X, Y)]$.
Questions
My first question is whether $g$ is at all well-defined: why are we allowed to "freeze" one variable here?
Then, I also cannot convince myself that it $g$ is at all measurable I feel it follows from the Monotone Class Theorem and the fact but cannot formalize it.
Ultimately, any hints about proving the statement would be appreciated -- I do not know where to even start.