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 $G$ 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:

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

Monotone class theorem for sets

Monotone class theorem for sets — Let $G$ 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:

If $A\in {\mathcal {A}}$ then $\mathbf {1} _{A}\in {\mathcal {H}}$ where $\mathbf {1} _{A}$ denotes the indicator function of $A.$
If $f,g\in {\mathcal {H}}$ and $c\in \mathbb {R}$ then $f+g$ and $cf\in {\mathcal {H}}.$
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 𝜎-algebra generated by ${\mathcal {A}}.$

Proof

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

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, $\sigma ({\mathcal {A}})\subseteq {\mathcal {G}}.$ 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 𝜎-ring of $G.$

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.

Citations

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.

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