Is the product of Borel spaces a Borel space?

Let $(S_i,\mathbf{S}_i)$ be a sequence of Borel spaces, i.e. such that for all $i$ there is a 1-1 bimeasurable map $\varphi_i:S_i\to T_i$, where $T_i$ is a Borel subset of [0,1].

Is $\prod_{i=1}^n S_i$, equipped with the product $\sigma$-algebra, a Borel space?

What about $\prod_{i=1}^\infty S_i$, with the $\sigma$-algebra $\mathbf{A}$, where $\mathbf{A}$ is the smallest $\sigma$-algebra containing elements of the form $\prod T_i$, where $T_i$=$S_i$ for all but finitely many $i$'s?

Thank you.

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Hint: $[0,1]^n$ and $[0,1]^\infty$ are Borel spaces. –  Nate Eldredge Jul 9 '11 at 18:44