- 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!