$\Omega = \{a,b,c\}$
$\mathcal{C}=\{\{a\},\{b\}\} \subset \mathcal{P}(\Omega)$
- What is the $\sigma$-algebra and the $\lambda$-system generated by the class $\mathcal{C}$ described above?
- Will the $\lambda$-system contain the empty set $\emptyset$?If yes, then please give an example of a class of sets $\mathcal{D}$ which is not a $\pi$-system and $\sigma\langle\mathcal{D}\rangle \neq\lambda \langle\mathcal{D}\rangle$.
I was working on the Dynkin $\pi$-$\lambda$ theorem, and was getting $\sigma\langle\mathcal{D}\rangle =\lambda \langle\mathcal{D}\rangle$ for every non-$\pi$-system $\mathcal{D}$.
Best Answer
Recall that a $\lambda$-system is closed under complements and disjoint countable unions, while a $\sigma$-algebra is closed under complements and arbitrary countable unions. So every $\sigma$-algebra is a $\lambda$-system, but not vice versa.
In this case, the $\lambda$-system generated by $\mathcal C$ and the $\sigma$-algebra generated by $\mathcal C$ are both $2^\Omega$. For an example of where the $\lambda$-system and $\sigma$-algebra are different, consider $\Omega=\{0,1,2,3\}$ and
$$ \left\{\varnothing, \{0,2\}, \{0,3\}, \{1,2\}, \{1, 3\}, \{0,1,2,3\} \right\},$$ which is the $\lambda$-system generated $$\mathcal C=\{\{0,2\},\{0,3\}\}.$$ The $\sigma$-algebra generated by this class would again be $2^\Omega$.