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
[edit]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:
- 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 - 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
[edit]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
[edit]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:
- 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.} - If 👁 {\displaystyle f,g\in {\mathcal {H}}}
and 👁 {\displaystyle c\in \mathbb {R} }
then 👁 {\displaystyle f+g}
and 👁 {\displaystyle cf\in {\mathcal {H}}.} - 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
[edit]The following argument originates in Rick Durrett's Probability: Theory and Examples.[1]
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
[edit]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
[edit]- 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
Citations
[edit]- ^ Durrett, Rick (2010). Probability: Theory and Examples (4th ed.). Cambridge University Press. p. 276. ISBN 978-0521765398.
References
[edit]- 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.