Let be $X$ a metric compact space and $(G,+)$ a topological compact abelian group. Let be $\mathcal{A}$ the Borel $\sigma$-algebra of $X$ and $\mathcal{B}$ the Borel $\sigma$-algebra of $G$. Consider in $X\times G$ the product $\sigma$-algebra.
My Question: I have a Borelian $A\times G$ of $X\times G.$ I want to show is that the set $ A $ must be necessarily a borel set of $X$