Relation between field of subsets, monotone class, $\lambda$ and $\pi$ systems 
*

*For a field of subsets, the monotone class generated by the field is
the same as the $\sigma$ algebra generated by the field.

*Since a class of subsets is both a monotone class and a field of
subsets if and only if it is a $\sigma$ algebra, it follows that, 
for a class of subsets, taking monotone class "closure" doesn't
change being a field.

*Any $\lambda$ system containing the $\pi$ system contains the $\sigma$ algebra generated by the $\pi$ system. (Dynkin's π-λ Theorem)
Questions:


*

*Are the statements true when switching "filed of subsets" and
"monotone class"?
I.e. 
For a monotone class, is the field generated by the monotone class
same as the $\sigma$ algebra generated by the monotone class?
For a class of subsets, does taking monotone class "closure" change being a field?
For a monotone class, does any field containing the monotone class contains the $\sigma$ algebra generated by the monotone class? (Analogous to Dynkin's $π-λ$ Theorem)

*Are the statements true when replacing "filed of subsets" with
"$\lambda$ system" and "monotone class" with "$\pi$ system"?
I.e. 
For a $\pi$ system, is the $\lambda$ system generated by the $\pi$
system same as the $\sigma$ algebra generated by $\pi$ system?
For a class of subsets, does taking $\pi$ system "closure" change
being a $\lambda$ system?

*Are the statements true when switching "$\lambda$ system" and "$\pi$
system" in the previous part?
I.e. 
For a $\lambda$ system, is the $\pi$ system generated by the
$\lambda$ system same as the $\sigma$ algebra generated by $\lambda$
system?
For a class of subsets, does taking $\lambda$ system "closure"
change being a $\pi$ system?
For a $\lambda$ system, does any $\pi$ system containing the $\lambda$ system contains the $\sigma$ algebra generated by the $\lambda$ system? (Analogous to Dynkin's $π-λ$ Theorem)
Thanks and regards!
 A: The following example will be useful: 
Let $P$ be a countably infinite set of vectors in $\mathbb{R}^2_+$ with norm $1$. The family $$\{\mathbb{R}^2,\emptyset\}\cup\big\{\{x\in\mathbb{R}^2\}:px\geq 0:p\in P\}\big\}\cup\big\{\{x\in\mathbb{R}^2\}:px< 0:p\in P\}\big\}$$ is both a countably infinite $\lambda$-system and monotone class on $\mathbb{R}^2$.


*

*Take some countably infinite monotone class. The field generated by a countably infinity family is countably infinite, but no $\sigma$-algebra is countably infinite. The field generated thus also contains the monotone class but not the $\sigma$-algebra generated.  Since the monotone class generated by a field is a $\sigma$-algebra, it is trivially a field. 

*Your first question is directly answered by the $\pi-\lambda$-Theorem of Dynkin. For the second, take a countably infinite $\lambda$-system. The $\pi$-system generated by $\lambda$ is countably infinite too. If it were still a $\lambda$-system, it would be a $\sigma$-algebra.

*Your first and third question can again be answered by a countability argument, the second by the $\pi-\lambda$-Theorem of Dynkin.
