---- > [!definition] Definition. ([[monotone class]]) > A **monotone class** is a collection $\mathscr{M}$ of sets that is stable under [[countably infinite|countable]] [[monotonic map|increasing]] unions and under [[countably infinite|countable]] [[monotonic map|decreasing]] intersections, that is: > 1. If $E_{1} \subset E_{2} \subset \cdots$ is an increasing sequence of sets in $\mathscr{M}$, then $\bigcup_{k=1}^{\infty} E_{k} \in \mathscr{M}$; > 2. If $E_{1} \supset E_{2} \supset \cdots$ is a decreasing sequence of sets in $\mathscr{M}$ , then $\bigcap_{k=1}^{\infty}E_{k} \in \mathscr{M}$. > > ![[class methods in measure theory.canvas|class methods in measure theory]] > [!basicexample] > - Clearly every [[σ-algebra]] is a monotone class. > - However, some monotone classes are not even stable under finite unions (i.e. are not [[algebra of sets|algebras]]). For example, the collection $\mathscr{M}$ of all [[interval|intervals]] of $\mathbb{R}$ is certainly a monotone class, but because $\mathscr{M}$ is not closed under finite unions (nor complementation) it is not an [[algebra of sets|algebra]] (nor a [[σ-algebra]]). > - The intersection of all monotone classes containing some set collection $\mathscr{A}$ is again a monotone class containing $\mathscr{A}$, called the **monotone class generated by $\mathscr{A}$**. > - [[Dynkin system|Dynkin systems]] are monotone classes with added conditions regarding complements ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```