Definition:Monotone Class

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Definition

Let $X$ be a set, and let $\mathcal P \left({X}\right)$ be its power set.

Let $\mathcal M \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$.


Then $\mathcal M$ is said to be a monotone class (on $X$) if and only if for every countable, nonempty, index set $I$, it holds that:

$\displaystyle \left({A_i}\right)_{i \in I} \in \mathcal M \implies \bigcup_{i \mathop \in I} A_i \in \mathcal M$
$\displaystyle \left({A_i}\right)_{i \in I} \in \mathcal M \implies \bigcap_{i \mathop \in I} A_i \in \mathcal M$

that is, if and only if $\mathcal M$ is closed under countable unions and intersections.


Also defined as

Some sources stipulate only that above closure properties should hold for $I = \N$.

This definition is equivalent, as proved on Equivalence of Definitions of Monotone Class.


Also see

  • Results about monotone classes can be found here.


Sources