# Is every sigma algebra generated by some subclass.

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 $\cal{C}?$

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.

• You had better put some condition on $\mathcal C$, otherwise just take $\mathcal A = \mathcal C$. – treble May 20 '18 at 19:47
• Thanks for the warning. When I ask the question I have the condition ${\cal C}$ strictly contained in ${\cal A}$ in mind – Eduardo May 20 '18 at 19:55

If you take $$\mathcal{C}=\mathcal A$$, then you'll have a set that generates $$\mathcal A$$. If you want a smaller set, take $$\mathcal{C}=\mathcal{A}\setminus\{\emptyset\}$$.