# Projections of sets in product sigma algebras over a countable set

I've been looking through some posts about how projections of sets from product $\sigma$-algebras of two Borel $\sigma$-algebras onto their component spaces are not necessarily measurable which apparently leads into the rabbit hole of descriptive set theory.

My problem, though, is a little different and I thought perhaps someone had an idea of how to tackle it seeing as how I am not really knowledgeable when it comes to descriptive set theory.

What I have is a kind of Polish space $\mathcal{X}$ (complete, separable metric space), a countable (or even finite) set $\mathcal{A}$ (being polish by virtue of its discrete metric if I am not mistaken) and their respective Borel $\sigma$-algebras $\mathcal{B}({\mathcal{X}})$ and $\mathcal{B}(\mathcal{A})=2^{\mathcal{A}}$ since $\mathcal{A}$ is countable.

What I am wondering now is, if I look at some set from their product $S \in \mathcal{B}({\mathcal{X}}) \otimes \mathcal{B}(\mathcal{A})$, does the setup now give us that $$\pi_{\mathcal{X}}(S) := \{ x \in \mathcal{X} | \, \exists\, a \in \mathcal{A}: \, (x,a) \in S \}$$

What I have tried to use is that since $\mathcal{B}(\mathcal{A})=2^{\mathcal{A}}$ and countable, we can write $S$ as

$$S = \bigcup_{a \in \mathcal{A}} S_a \times \{a \}$$ where $S_a = \{ x \in \mathcal{X}|\, (x,a) \in S \}$ hoping I could somehow conclude that $S_a \in \mathcal{B}(\mathcal{X})$ as intuitively, pairing Borel sets with countably (or even finitely) many point sets and using that as a generator for $\mathcal{B}({\mathcal{X}}) \otimes \mathcal{B}(\mathcal{A})$ shouldn't all of a sudden provide us with sets whose projections are non-Borel.

However, I cannot seem to properly prove this statement and perhaps it is false to begin with. I thought someone here could give some insights.

Thanks!

Your argument goes through. The set $S_a = \{ x \in \mathcal{X}|\, (x,a) \in S \}$ is known as the $a$-section, and it is generally measurable for the product $\sigma$-algebra (the smallest $\sigma$-algebra generated by rectangles).