Let $\Omega$ a non empty set and consider and $\cal{A}$ a sigma algebra in $\Omega$. Is there always a set $\cal{C}\subset\mathscr{P}(\Omega)$, $\cal{C}$ strictly contained in $\cal{A}$ such that $\cal{A}$ is generated by $\cal{C}?$
For example, is the Lebesgue measure generated by some $\cal{C}?$
Obs. https://mathoverflow.net/questions/87838/is-every-sigma-algebra-the-borel-algebra-of-a-topology
I have already seen the discussion in the above link, but I'm not asking noting special about the class $\cal{C}$.
Also in the above case the answer is quite sophisticated. I'm wondering if in the question I ask the answer is more simple.
If you take $\mathcal{C}=\mathcal A$, then you'll have a set that generates $\mathcal A$. If you want a smaller set, take $\mathcal{C}=\mathcal{A}\setminus\{\emptyset\}$.