Let $X=X_{1}\times X_{2}$ is locally compact space, and define $$E=\{E_{1}\times E_{2}\mid E_{i}\text{ is a Borel set in }X_{i}\;,\text{ for}\; i=1,2\}$$ Now why the Baire sets of $X$ are in the $\sigma$-algebra generated by $E$? Of course that every Baire set is Borel too so all Baires of $X_{i}$ is a Borel of it too. Locally Compact: every point has a neighborhood with compact closure.

Baires: The smallest σ -algebra of subsets of $X$ such that each function in $C_{C}(X)$ is measurable with respect to that.