Monotone class theorem

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

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

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 ${\displaystyle \mathbf {1} _{A}\in {\mathcal {H}}.}$ 
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 sigma-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, ${\displaystyle \sigma ({\mathcal {A}})\subset {\mathcal {G}}.}$  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 sigma-ring of ${\displaystyle G.}$ 

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

Citations

1. Durrett, Rick (2010). Probability: Theory and Examples (4th ed.). Cambridge University Press. p. 276. ISBN 978-0521765398.

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.