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) $M$ of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means $M$ has the following properties:

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 $G$; that is, $\sigma (G)=M(G).$

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
• π-system – Family of sets closed under intersection
• Dynkin system – Family closed under complements and countable disjoint unions

References

• Durrett, Richard (2019). Probability: Theory and Examples (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. Vol. 49 (5th ed.). Cambridge New York, NY: Cambridge University Press. ISBN 978-1-108-47368-2. OCLC 1100115281. Retrieved November 5, 2020.

Families ${\mathcal {F}}$ of sets over $\Omega$
Is necessarily true of ${\mathcal {F}}\colon$ or, is ${\mathcal {F}}$ closed under:	Directed by $\,\supseteq$	$A\cap B$	$A\cup B$	$B\setminus A$	$\Omega \setminus A$	$A_{1}\cap A_{2}\cap \cdots$	$A_{1}\cup A_{2}\cup \cdots$	$\Omega \in {\mathcal {F}}$	$\varnothing \in {\mathcal {F}}$	F.I.P.
π-system
Semiring										Never
Semialgebra (Semifield)										Never
Monotone class						only if $A_{i}\searrow$	only if $A_{i}\nearrow$
𝜆-system (Dynkin System)				only if $A\subseteq B$			only if $A_{i}\nearrow$ or they are disjoint			Never
Ring (Order theory)
Ring (Measure theory)										Never
δ-Ring										Never
𝜎-Ring										Never
Algebra (Field)										Never
𝜎-Algebra (𝜎-Field)										Never
Dual ideal
Filter				Never	Never				$\varnothing \not \in {\mathcal {F}}$
Prefilter (Filter base)				Never	Never				$\varnothing \not \in {\mathcal {F}}$
Filter subbase				Never	Never				$\varnothing \not \in {\mathcal {F}}$
Open Topology							(even arbitrary unions)			Never
Closed Topology						(even arbitrary intersections)				Never
Is necessarily true of ${\mathcal {F}}\colon$ or, is ${\mathcal {F}}$ closed under:	directed downward	finite intersections	finite unions	relative complements	complements in $\Omega$	countable intersections	countable unions	contains $\Omega$	contains $\varnothing$	Finite Intersection Property
Additionally, a semiring is a π-system where every complement $B\setminus A$ is equal to a finite disjoint union of sets in ${\mathcal {F}}.$ A semialgebra is a semiring that contains $\Omega .$ $A,B,A_{1},A_{2},\ldots$ are arbitrary elements of ${\mathcal {F}}$ and it is assumed that ${\mathcal {F}}\neq \varnothing .$