monotone class

Monotone ClassFernando Sanz Gámiz

Definition.

A collection $ℳ$ of subsets of $\mathrm{\Omega }$ is a monotone classif it is closed under increasing and decreasing sequences of sets,i.e.

1. $A_1\subseteq A_2\subseteq A_3,\mathrm{\dots }\in ℳ\Rightarrow \bigcup A_n\in ℳ$

2. $A_1\supseteq A_2\supseteq A_3,\mathrm{\dots }\in ℳ\Rightarrow \bigcap A_n\in ℳ$