query: $\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated iff there is a random variable $X$ such that $\mathcal{G} = \sigma(X)$.
positive_passages.docid: 146951
positive_passages.text: Every countably generated sigma algebra is generated by a random variable
negative_passages.docid: 46056
negative_passages.text: If $\mathcal{E}$ are subsets of $X$ and $A \subset X$. Show that $\mathcal{A}\mathcal{(E \cap}$ A)=$\mathcal{A}\mathcal{(E)} \cap$ A
negative_passages.docid: 135548
negative_passages.text: Every section of a measurable set is measurable? (in the product sigma-algebra)
negative_passages.docid: 145428
negative_passages.text: Set $A$ is countably infinite if and only if there exists a bijection $f: \mathbb{N} \rightarrow A$
negative_passages.docid: 143732
negative_passages.text: Prove random variable $Y_0$ and $\sigma$-algebra $\mathscr{R}$ are independent.
negative_passages.docid: 117133
negative_passages.text: Is there an infinite countable $\sigma$-algebra on an uncountable set
negative_passages.docid: 136094
negative_passages.text: $G_\delta$ singletons in compact Hausdorff and first countability
negative_passages.docid: 23907
negative_passages.text: $Seq (\mathbb{N})$ of all finite sequences of elements of $\mathbb{N}$ is countable.
negative_passages.docid: 63860
negative_passages.text: Measure of Dependence Between Two $\sigma$-algebras
negative_passages.docid: 64118
negative_passages.text: Number of elements in a finite $\sigma$-algebra
negative_passages.docid: 147486
negative_passages.text: Cartesian product of two $\sigma$-algebras is generally not a $\sigma$-algebra?
negative_passages.docid: 10702
negative_passages.text: Show that $\left\{A\subset\Omega: A~\text{ is countable or }A^C\text{ is countable}\right\}$ is a $\sigma$-algebra
negative_passages.docid: 98897
negative_passages.text: Proving that the set of $\nu$ measurable sets $M\subseteq P(X)$ is an algebra
negative_passages.docid: 16413
negative_passages.text: Proving existence of surjective $f:\mathbb{N} \rightarrow A$ implies $A$ is at most countable.
negative_passages.docid: 34998
negative_passages.text: How to prove that $\mathbb{Q}$ ( the rationals) is a countable set
negative_passages.docid: 103427
negative_passages.text: When is $\operatorname{Hom}_A(M,N)$ finitely generated?
negative_passages.docid: 127522
negative_passages.text: Space $\mathcal{L}^p(X, \Sigma, \mu)$ is separable iff $(\Sigma, \rho_\Delta)$ is separable