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Let $X=X_{1}\times X_{2}$ is locally compact space, and define $$E=\{E_{1}\times E_{2}\;|\; E_{i}\; \text{is a Borel set in}\; X_{i}\; ,\; \text{for}\; i=1,2\}$$ 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.

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  • $\begingroup$ I deleted a straightforward answer based on the Stone-Weierstrass theorem because apparently AmirHosein doesn't assume the Hausdorff condition. Please specify what you mean by local compactness: a compact neighborhood for every point? a base of compact neighborhoods for every point? a base of closed compact neighborhoods for every point? What is your definition of the Baire $\sigma$-algebra? The one generated by the zero sets of continuous functions or the one generated by (closed?) compact $G_\delta$-sets? $\endgroup$ – Martin Jan 2 '13 at 22:49
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    $\begingroup$ Asked and answered on MO. $\endgroup$ – Martin Jan 5 '13 at 15:05
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(To kick it from the unanswered queue)

Original MO link: Baire sets of $X$ possess the required Cartesian product property (as mentioned in the comment)

@JosephVanName's solution:

The idea of the proof is to make use of compactness to show that sets of the form $f^{-1}(0)^{c}$ are countable unions of boxes in $E$.

For a proof, assume that $f:X^{2}\rightarrow[0,1]$ is a continuous function with compact support. Let $U=f^{-1}(0,1]$. Then since $f$ has compact support, the sets $f^{-1}[a,1]$ are compact for $a>0$, so the set $U$ is Lindelof. If $x\in U$, then by the definition of the product topology there are open sets $A_{1}\subseteq X_{1},A_{2}\subseteq X_{2}$ with $x\in A_{1}\times A_{2}\subseteq U$. If we set $A_{x}=A_{1}\times A_{2}$, then $A_{x}\in E$. Furthermore, by the Lindelof property, there is some $\{x_{n}|n\in\mathbb{N}\}\subseteq U$ with $U=\bigcup_{n}A_{x_{n}}$. Therefore, we have $U$ be in the $\sigma$-algebra generated by $E$ as well. Therefore since each the $\sigma$-algebra generated by $E$ contains each cozero set of compact support, the $\sigma$-algebra generated by $E$ contains each Baire set.

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