We define a class of subsets of $\Omega$ to be a Monotone Class $\mathcal{M}$ iff it has the following two properties:

$(1)$ if $A_i \in \mathcal{M}$ and $A_1 \subset A_2 \subset \dots $ then $\bigcup_{i\ge 1} A_i \in \mathcal{M}$

$(2)$ if $A_i \in \mathcal{M}$ and $A_1 \supset A_2 \supset \dots$ then $\bigcap_{i\ge 1} A_i \in \mathcal{M}$

But, doesn't every set satisfies these properties as if $A_i \in \mathcal{M}$ and $A_1 \subset A_2 \subset ... \subset A_n$ then $\bigcup_{i\ge 1}^{i=n} A_i = A_n \in \mathcal{M}$. Similarly, if $A_i \in \mathcal{M}$ and $A_1 \supset A_2 \supset ... \supset A_n$ then $\bigcap_{i\ge 1}^{i=n} A_i = A_n \in \mathcal{M}$.

Is there a class of subsets which isn't Monotone Class?

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    $\begingroup$ The idea is that the sequence of $A_n$'s can be infinite. That's what the $\dots$ at the end is supposed to indicate. $\endgroup$ – bof Jan 13 '16 at 7:49
  • $\begingroup$ @bof but then there should be a class which isn't monotone class. Why else would we give it a special name monotone then? $\endgroup$ – vivkul Jan 13 '16 at 7:55
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    $\begingroup$ There are lots of classes which aren't monotone. If $\Omega=\mathbb R$, examples of non-monotone classes include: the class of all finite subsets of $\mathbb R$; the class of all bounded subsets; the class of all open subsets; the class of all closed subsets, and many others. $\endgroup$ – bof Jan 13 '16 at 8:06

If $\Omega$ is a finite set, then every class of subsets of $\Omega$ is monotone.

If $\Omega$ is an infinite set, then the class $\mathcal M$ of all nonempty subsets of $\Omega$ is a non-monotone class. To see that $\mathcal M$ is not monotone, choose an infinite sequence $a_1,a_2,\dots,a_n,\dots$ of distinct elements of $\Omega,$ and for each $i\in\mathbb N$ define $A_i=\bigcup_{n=i}^\infty\{a_n\}.$ Then $A_i\in\mathcal M$ for each $i$, and $A_1\supset A_2\supset\cdots\supset A_i\supset\cdots,$ and $\bigcap_{i=1}^\infty A_i=\emptyset\notin\mathcal M.$

  • $\begingroup$ a very nice explanation. Thank you. $\endgroup$ – vivkul Jan 13 '16 at 9:50

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