Definition:Monotone Class

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Definition

Let $X$ be a set, and let $\powerset X$ be its power set.

Let $\MM \subseteq \powerset X$ be a collection of subsets of $X$.


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

$\displaystyle \family {A_i}_{i \mathop \in I} \in \MM \implies \bigcup_{i \mathop \in I} A_i \in \MM$
$\displaystyle \family {A_i}_{i \mathop \in I} \in \MM \implies \bigcap_{i \mathop \in I} A_i \in \MM$

that is, if and only if $\MM$ is closed under countable unions and countable 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