Is every sigma-algebra generated by some random variable?

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)

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).