---
tag: 2T33SA540E
kind: statement
role: theorem
name: Monotone class theorem
msc: [28A05]
uses: [PFMGZ7988V, 0SV8K0TDGY]
about: [PFMGZ7988V, 0SV8K0TDGY]
sources:
  - url: https://en.wikipedia.org/wiki/Monotone_class_theorem
same_as:
  wikidata: Q242045
  wikipedia: Monotone class theorem
---

## Statement

Let $X$ be a set.
A *monotone class* on $X$ is a family $\mathcal{M}$ of subsets of $X$ with these two properties.
If $A_1 \subseteq A_2 \subseteq \cdots$ are in $\mathcal{M}$, then $\bigcup_{n=1}^{\infty} A_n \in \mathcal{M}$.
If $A_1 \supseteq A_2 \supseteq \cdots$ are in $\mathcal{M}$, then $\bigcap_{n=1}^{\infty} A_n \in \mathcal{M}$.

Let $\mathcal{A}$ be an [[algebra of sets|PFMGZ7988V]] on $X$.
Then the smallest monotone class on $X$ that contains $\mathcal{A}$ is equal to the [[σ-algebra|0SV8K0TDGY]] $\sigma(\mathcal{A})$ generated by $\mathcal{A}$.

## Remarks

Every σ-algebra is a monotone class, so the smallest monotone class $m(\mathcal{A})$ is contained in $\sigma(\mathcal{A})$.
For the other inclusion, one shows that $m(\mathcal{A})$ is an algebra.
For fixed $A$, the sets $B$ with $A \cup B$, $A \setminus B$ and $B \setminus A$ in $m(\mathcal{A})$ form a monotone class; this gives closure first for $A \in \mathcal{A}$, then for $A \in m(\mathcal{A})$.
An algebra that is a monotone class is a σ-algebra, since a countable union is the increasing union of finite unions.

As an application, let $\mu$ and $\nu$ be finite measures on $(X, \sigma(\mathcal{A}))$ with $\mu(X) = \nu(X)$ that agree on $\mathcal{A}$.
The sets on which they agree form a monotone class, by continuity of measures, so $\mu = \nu$ on $\sigma(\mathcal{A})$.

The hypothesis that $\mathcal{A}$ is an algebra is needed.
On $X = \set{1, 2}$, the family $\set{\varnothing, \set{1}}$ is already a monotone class, but the σ-algebra it generates also contains $\set{2}$.

## Explanations

### L1 human

A monotone class is a family of sets that is stable under limits of growing sequences and of shrinking sequences.
The theorem says: start from an algebra of sets, and close it under such limits only.
Then you already get every set of the generated σ-algebra.
So a property that survives monotone limits and holds on the algebra holds on the whole σ-algebra.

### L2 human

The theorem is a tool to extend statements from simple sets to all measurable sets.
It is used to prove uniqueness of extended measures and to set up product measures and the Fubini–Tonelli theorem.
Dynkin's π-λ theorem is a variant in which the algebra is replaced by a π-system and the monotone class by a Dynkin system.
There is also a version for classes of functions, used in probability theory.