# Monotone class theorem

In measure theory and probability, the monotone class theorem connects monotone classes and sigma-algebras. The theorem says that the smallest monotone class containing an algebra of sets G is precisely the smallest σ-algebra containing G. It is used as a type of transfinite induction to prove many other theorems, such as Fubini’s theorem.

Contents

1 Definition of a monotone class
2 Monotone class theorem for sets

2.1 Statement

3 Monotone class theorem for functions

3.1 Statement
3.2 Proof

4 Results and Applications
5 References

Definition of a monotone class
A monotone class in a set

R

{\displaystyle R}

is a collection

M

{\displaystyle M}

of subsets of

R

{\displaystyle R}

which contains

R

{\displaystyle R}

and is closed under countable monotone unions and intersections, i.e. if

A

i

M

{\displaystyle A_{i}\in M}

and

A

1

A

2

{\displaystyle A_{1}\subset A_{2}\subset \ldots }

then

i
=
1

A

i

M

{\displaystyle \cup _{i=1}^{\infty }A_{i}\in M}

, and similarly for intersections of decreasing sequences of sets.
Monotone class theorem for sets
Statement
Let G be an algebra of sets and define M(G) to be the smallest monotone class containing G. Then M(G) is precisely the σ-algebra generated by G, i.e. σ(G) = M(G)
Monotone class theorem for functions
Statement
Let

A

{\displaystyle {\mathcal {A}}}

be a π-system that contains

Ω

{\displaystyle \Omega \,}

and let

H

{\displaystyle {\mathcal {H}}}

be a collection of functions from