Monotone Class Theorem

Author

John Robin Inston

Published

September 25, 2026

1 Intuition

The monotone class theorem connects monotone classes and \(\sigma\)-algebras. It states that the smallest monotone class containing an algebra of sets \(G\) is precisely the [[generated-sigma-algebra|smallest \(\sigma\)-algebra]] containing \(G\). It acts as a type of transfinite induction: to prove a property for all measurable sets (or functions) it suffices to prove it on a simple generating class and check closure under monotone limits. It is used to prove many results, including Fubini’s theorem.

2 Monotone Class Theorem for Sets

A monotone class is a family \(M\) of sets that is closed under countable monotone unions and under countable monotone intersections. Explicitly:

  1. if \(A_{1}, A_{2}, \dots \in M\) and \(A_1\subseteq A_{2} \subseteq \dots\) then \(\bigcup_{i=1}^\infty A_{i}\in M\);
  2. if \(B_{1}, B_{2}, \dots \in M\) and \(B_1\supseteq B_{2} \supseteq \dots\) then \(\bigcap_{i=1}^\infty B_{i}\in M\).

Let \(G\) be an algebra of sets and let \(M(G)\) be the smallest monotone class containing \(G\). Then \(M(G)=\sigma(G)\), the \(\sigma\)-algebra generated by \(G\).

Proof. Every \(\sigma\)-algebra is a monotone class, so \(M(G)\subseteq\sigma(G)\). For the converse it suffices to show \(M:=M(G)\) is an algebra, since a monotone class that is an algebra is a \(\sigma\)-algebra (a countable union is the increasing limit of finite unions).

  • Complements. \(\{A: A^c\in M\}\) is a monotone class containing \(G\), hence contains \(M\).
  • Intersections. For \(A\in G\), \(\{B: A\cap B\in M\}\) is a monotone class containing \(G\), hence contains \(M\). So \(A\cap B\in M\) for \(A\in G\), \(B\in M\). Now fix \(B\in M\): \(\{A: A\cap B\in M\}\) is a monotone class containing \(G\) by the previous step, hence contains \(M\).

Thus \(M\) is closed under complements and finite intersections, and contains \(\Omega\), so it is an algebra.

3 Monotone Class Theorem for Functions

Let \(\mathcal{A}\) be a [[pi-system|\(\pi\)-system]] that contains \(\Omega\) and let \(\mathcal{H}\) be a collection of functions from \(\Omega\) to \(\mathbb{R}\) with the following properties:

  1. if \(A\in\mathcal{A}\) then \(\mathbb{1}_{A}\in \mathcal{H}\);
  2. if \(f,g \in \mathcal{H}\) and \(c \in \mathbb{R}\) then \(f+g\in \mathcal{H}\) and \(cf \in \mathcal{H}\); and
  3. if \(f_{n}\in \mathcal{H}\) is a sequence of non-negative functions that increase to a bounded function \(f\) then \(f \in\mathcal{H}\).

Then \(\mathcal{H}\) contains all bounded functions that are measurable with respect to \(\sigma(\mathcal{A})\), the [[generated-sigma-algebra|\(\sigma\)-algebra generated]] by \(\mathcal{A}\).

4 References

  • R. Durrett, Probability: Theory and Examples, Cambridge University Press.
  • P. Billingsley, Probability and Measure, Wiley.
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