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.

Definition of a monotone class

A monotone class is a family (i.e. class) M that is closed under countable monotone unions and also countable monotone intersections. Explicitly, this means:

1. if $A_{1},A_{2},\ldots \in M$ and $A_{1}\subseteq A_{2}\subseteq \cdots$ then ${\textstyle \bigcup _{i=1}^{\infty }A_{i}\in M,}$ and
2. if $B_{1},B_{2},\ldots \in M$ and $B_{1}\supseteq B_{2}\supseteq \cdots$ then ${\textstyle \bigcap _{i=1}^{\infty }B_{i}\in M.}$

Monotone class theorem for sets

Monotone class theorem for sets—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

Monotone class theorem for functions—Let ${\mathcal {A}}$ be a π-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 $\mathbf {1} _{A}\in {\mathcal {H}}.$
2. If $f,g\in {\mathcal {H}}$ and $c\in \mathbb {R}$ then $f+g$ and $cf\in {\mathcal {H}}.$
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 $\sigma ({\mathcal {A}}),$ which is the sigma-algebra generated by ${\mathcal {A}}.$

Proof

The following argument originates in Rick Durrett's Probability: Theory and Examples.

Proof

The assumption $\Omega \,\in {\mathcal {A}},$ (2), and (3) imply that ${\mathcal {G}}=\left\{A:\mathbf {1} _{A}\in {\mathcal {H}}\right\}$ is a λ-system. By (1) and the π−λ theorem, $\sigma ({\mathcal {A}})\subset {\mathcal {G}}.$ Statement (2) implies that ${\mathcal {H}}$ contains all simple functions, and then (3) implies that ${\mathcal {H}}$ contains all bounded functions measurable with respect to $\sigma ({\mathcal {A}}).$

Results and applications

As a corollary, if G is a ring of sets, then the smallest monotone class containing it coincides with the sigma-ring of G.

By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a sigma-algebra.

The monotone class theorem for functions can be a powerful tool that allows statements about particularly simple classes of functions to be generalized to arbitrary bounded and measurable functions.

See also

• π-λ theorem – Family closed under complements and countable disjoint unionsPages displaying short descriptions of redirect targets
• π-system – Family of sets closed under intersection
• Dynkin system – Family closed under complements and countable disjoint unions