I am reading [this introduction to enlargement of filtration](http://www.minet.uni-jena.de/Marie-Curie-ITN/EoF/talks/jeanblanc_introduction.pdf) and at the beginning of section 2.4 there is a claim that I cannot justify but seems like it should be well known. The author claims that if $\mathcal{F}\_t$ is the natural filtration of a Brownian Motion, then since $\mathcal{F}\_t$ is separable it follows that $\mathcal{F}\_\infty$ is generated by a real random variable. I assume that separable means that there is a countable collection of random variables which generates $\mathcal{F}\_t$,though I have not been able to find a definition for the term 'separable' in this context. In any case, I wonder if someone could help me with a proof (or a reference for one) of this result.



If $\mathcal{F}_t$ is separable why is $\mathcal{F}_\infty$ generated by a random variable?
Let $(\Omega,\mathcal B,\mu)$ be a probability space and $\mathcal A$ a sub-sigma-algebra of $\mathcal B$. The following statements are equivalent: 


1. $\mathcal A$ admits a countable set of generators.
2. There is a bounded random variable $X\colon \Omega\to \Bbb R$ such that $\sigma(X)=\mathcal A$.
3. There are countably many bounded random variables $X\_k\colon \Omega\to \Bbb R$ such that $\sigma(X\_k,k\in\Bbb N)=\mathcal A$.


Indeed, if $1.$ is satisfied, let $\(A\_k,k\in\Bbb N\)\subset 2^\Omega$ generating $X$. Then define $X:=\sum\_{k=1}^{+\infty}3^{-k}\chi\_{A\_k}$. Then $X^{-1}(3^{-k})=A\_k$. 


$2.\Rightarrow 3.$ is obvious taking $X\_k=X$. 


$3.\Rightarrow 1.$ Let $S\_j$ be countably many sets generating Borel sigma-algebra, for example open intervals with rational endpoints. The (countable) collection $(X\_k^{-1}(S\_j),(k,j)\in\Bbb N^2)$ generated $\mathcal A$.


