# Definition:Monotone Class

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## Contents

## 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

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $\S 3$: Problem $11$