Monotone class theorem

In measure theory and probability, the monotone class theorem connects monotone classes and 𝜎-algebras. The theorem says that the smallest monotone class containing an algebra of sets ${\displaystyle G}$ is precisely the smallest 𝜎-algebra containing ${\displaystyle G.}$ 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) ${\displaystyle M}$ of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means ${\displaystyle M}$ has the following properties:

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

Monotone class theorem for sets

Monotone class theorem for sets—Let ${\displaystyle G}$ be an algebra of sets and define ${\displaystyle M(G)}$ to be the smallest monotone class containing ${\displaystyle G.}$ Then ${\displaystyle M(G)}$ is precisely the 𝜎-algebra generated by ${\displaystyle G}$; that is ${\displaystyle \sigma (G)=M(G).}$

Monotone class theorem for functions

Monotone class theorem for functions—Let ${\displaystyle {\mathcal {A}}}$ be a π-system that contains ${\displaystyle \Omega \,}$ and let ${\displaystyle {\mathcal {H}}}$ be a collection of functions from ${\displaystyle \Omega }$ to ${\displaystyle \mathbb {R} }$ with the following properties:

1. If ${\displaystyle A\in {\mathcal {A}}}$ then ${\displaystyle \mathbf {1} _{A}\in {\mathcal {H}}}$ where ${\displaystyle \mathbf {1} _{A}}$ denotes the indicator function of ${\displaystyle A.}$
2. If ${\displaystyle f,g\in {\mathcal {H}}}$ and ${\displaystyle c\in \mathbb {R} }$ then ${\displaystyle f+g}$ and ${\displaystyle cf\in {\mathcal {H}}.}$
3. If ${\displaystyle f_{n}\in {\mathcal {H}}}$ is a sequence of non-negative functions that increase to a bounded function ${\displaystyle f}$ then ${\displaystyle f\in {\mathcal {H}}.}$

Then ${\displaystyle {\mathcal {H}}}$ contains all bounded functions that are measurable with respect to ${\displaystyle \sigma ({\mathcal {A}}),}$ which is the 𝜎-algebra generated by ${\displaystyle {\mathcal {A}}.}$

Proof

The following argument originates in Rick Durrett's Probability: Theory and Examples.^[1]

Proof

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

Results and applications

As a corollary, if ${\displaystyle G}$ is a ring of sets, then the smallest monotone class containing it coincides with the 𝜎-ring of ${\displaystyle G.}$

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

• Dynkin system – Family closed under complements and countable disjoint unions
• 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
• σ-algebra – Algebraic structure of set algebra