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Let $\mathcal{A}$ be a $\sigma$-algebra over $\Omega$. Is there a function $f:\Omega\rightarrow\mathbb{R}$ such that $\mathcal{A}=f^{-1}(\mathfrak{B(\mathbb{R})})$? ($\mathfrak{B(\mathbb{R})}$ being the Borel field on the real line)

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Not necessarily. The Borel $\sigma$-algebra is generated by a countable class of measurable sets, namely $\mathcal D:=\{(a,b),a,b\in\Bbb Q\}$. By the transfer property, $$\mathcal A=f^{-1}(\mathcal B(\Bbb R))=f^{-1}(\sigma(\mathcal D))=\sigma(f^{—1}(\mathcal D)),$$ so $\mathcal A$ is generated by a countable class.

It may be not the case, for example when $(\Omega,\mathcal A,\mu)=([0,1],2^{[0,1]},\delta_0)$ (no need to specify a measure).

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    Thanks. Could you indicate an example of a $\sigma$-algebra that cannot be generated by a countable sub-family?2012-12-30
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    @Evan: take the $\sigma$-algebra of all subsets of an uncountable set. More generally, a $\sigma$-algebra generated by a set of bounded cardinality has bounded cardinality.2012-12-30
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    @QiaochuYuan: You mean to say that every countably generated sigma algebra is countable?2012-12-30
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    A countably generated sigma-algebra has cardinal at most $\mathfrak c$. So, for example, the sigma-algebra of Lebesgue-measurable sets is not countably generated, since it has cardinal $2^{\mathfrak c}$.2012-12-30
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    @Evan: no, I mean to say that a $\sigma$-algebra generated by a set of cardinality at most $S$ itself has cardinality at most $f(S)$ for some function of $S$, not necessarily the identity.2012-12-30
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    @GEdgar: How can you tell the cardinality of the $\sigma$-algebra of Lebesgue-measurable sets?2012-12-31
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    @EvanAad: The Lebesgue measurable sets contain all the subsets of the Cantor set. Therefore it is at least $2^\frak c$; on the other hand it's a subset of $\mathcal P(\mathbb R)$, so its cardinality cannot extend $2^c$. It follows that the equality ensues.2012-12-31
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    @AsafKaragila: Nice. Thanks.2012-12-31
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    I'd like to add (to complement Davide's answer as well as Qiaochu's and DEdgar's comments) that the cardinality of a countably geberated $\sigma$-algebra is derived in [Folland's "Real Analysis"](http://www.amazon.com/Real-Analysis-Techniques-Applications-Mathematics/dp/0471317160/ref=sr_1_1?ie=UTF8&qid=1356933965&sr=8-1&keywords=folland+real+analysis), Note 1.2, pp. 40-41.2012-12-31
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    Equality **ensues** ... OK.2012-12-31