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  • 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:

  1. 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)

  2. 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?

  3. 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!

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1 Answer 1

up vote 1 down vote accepted

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 $\mathbb{R}^2$.

  1. 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.

  2. 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.

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

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Upvoted long time ago. Thanks! I wonder in the definition of the family of subsets, if $\big\{\{x\in\mathbb{R}^2\}:px\geq 0:p\in P\}\big\}$ mean $\big\{\{x\in\mathbb{R}^2\:px\geq 0\}_{p\in P}\big\}$? –  Tim Mar 11 '12 at 0:54

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