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If $$C_1 = \{A \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \cdots\ \vert A \text{ is an open set in }\mathbb{R}\}$$ and $$C_2 = \{B \times \mathbb{R} \times \mathbb{R}\times \mathbb{R} \times \cdots \vert B \in \mathbb{B}(\mathbb{R})\}$$ where $\mathbb{B}(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R}$. It is very clear that $\sigma(C_1) = C_2$, but I am unable to prove it explicitly.

Thanks, rjp

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    Does $\sigma(C_1)$ denote the Borel $\sigma$-algebra on $C_1$?2012-05-17
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    @Alex $\sigma(C_1)$ denotes the smallest $\sigma$-algebra containing $C_1$.2012-05-17
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    I think that "bottom-up" description of the smallest $\sigma$-algebra might be useful here. This is given e.g. in Halmos, [p.26](http://books.google.com/books?id=-Rz7q4jikxUC&pg=PA26) for $\sigma$-rings, it should be easy to modify this to $\sigma$-algebras. See also [this answer](http://math.stackexchange.com/a/54334/) for some general comments about top-down and bottom-up approach.2012-05-17
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    Top-down also works: Obviously $\sigma(C_1) \subseteq C_2$. Now let $D := \{B \in \mathbb B(\mathbb R) \mid B \times \prod_{\mathbb N}\mathbb R \in \sigma(C_1)\}$ and show that $D$ is a $\sigma$-algebra containig all open sets.2012-05-17
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    @martini: Thanks for the very nice answer. I think this is a very neat trick!2012-05-17
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    @martini That is indeed a simpler approach than what I suggested. Perhaps you could post it as an answer. [What should one do when one's question has been answered in the comments?](http://meta.math.stackexchange.com/questions/1148/what-should-one-do-when-ones-question-has-been-answered-in-the-comments)2012-05-17
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    @MartinSleziak I'll do.2012-05-17

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