Monotone class theorem

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 $\set{2}$.

Going deeper

For a student

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.