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 $A_{1}, A_{2}, \dots \in M$ and $A_{1} \subseteq A_{2} \subseteq \dots$ then $⋃_{i = 1}^{\infty} A_{i} \in M,$ and
if $B_{1}, B_{2}, \dots \in M$ and $B_{1} \supseteq B_{2} \supseteq \dots$ then $⋂_{i = 1}^{\infty} B_{i} \in M .$

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 $σ (G) = M (G) .$

Monotone class theorem for functions

Monotone class theorem for functions—Let $𝒜$ be a $π$ -system that contains $Ω$ and let $ℋ$ be a collection of functions from $Ω$ to $ℝ$ with the following properties:

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

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

Proof

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

Proof

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

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.

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