Monotone class theorem

In measure theory and probability, the monotone class theorem connects monotone classes and [[Sigma-algebra|Template:Sigma-algebra]]s. The theorem says that the smallest monotone class containing an algebra of sets $G$ is precisely the smallest [[Sigma-algebra|Template:Sigma-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 Template:Em 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 $A_1,A_2,\dots \in M$ and $A_1\subseteq A_2\subseteq \cdots$ then $\bigcup _{i=1}^{\mathrm{\infty }}{A}_i\in M,$ and
2. if $B_1,B_2,\dots \in M$ and $B_1\supseteq B_2\supseteq \cdots$ then $\bigcap _{i=1}^{\mathrm{\infty }}{B}_i\in M.$

Monotone class theorem for functions

Proof

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

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|Template: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|Template: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

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References

• Template:Durrett Probability Theory and Examples 5th Edition