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 is precisely the smallest 𝜎-algebra containing It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem.
A monotone class is a family (i.e. class) of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means has the following properties:
Monotone class theorem for sets — Let be an algebra of sets and define to be the smallest monotone class containing Then is precisely the 𝜎-algebra generated by ; that is
Monotone class theorem for functions — Let be a π-system that contains and let be a collection of functions from to with the following properties:
Then contains all bounded functions that are measurable with respect to which is the 𝜎-algebra generated by
The following argument originates in Rick Durrett's Probability: Theory and Examples. [1]
The assumption (2), and (3) imply that is a 𝜆-system. By (1) and the π−𝜆 theorem, Statement (2) implies that contains all simple functions, and then (3) implies that contains all bounded functions measurable with respect to
As a corollary, if is a ring of sets, then the smallest monotone class containing it coincides with the 𝜎-ring of
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.
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 is precisely the smallest 𝜎-algebra containing It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem.
A monotone class is a family (i.e. class) of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means has the following properties:
Monotone class theorem for sets — Let be an algebra of sets and define to be the smallest monotone class containing Then is precisely the 𝜎-algebra generated by ; that is
Monotone class theorem for functions — Let be a π-system that contains and let be a collection of functions from to with the following properties:
Then contains all bounded functions that are measurable with respect to which is the 𝜎-algebra generated by
The following argument originates in Rick Durrett's Probability: Theory and Examples. [1]
The assumption (2), and (3) imply that is a 𝜆-system. By (1) and the π−𝜆 theorem, Statement (2) implies that contains all simple functions, and then (3) implies that contains all bounded functions measurable with respect to
As a corollary, if is a ring of sets, then the smallest monotone class containing it coincides with the 𝜎-ring of
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.