measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...

and

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

, the monotone class theorem connects monotone classes and sigma-algebras. The theorem says that the smallest monotone class containing an

algebra of sets In mathematics, the algebra of sets, not to be confused with the mathematical structure of ''an'' algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the ...

G

is precisely the smallest -algebra containing

G.

It is used as a type of

transfinite induction Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for a ...

to prove many other theorems, such as

Fubini's theorem In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...

Definition of a monotone class

A ' is a

family Family (from la, familia) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its ...

(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

\bigcup_^ A_i \in M,

and # if

B_1, B_2, \ldots \in M

and

B_1 \supseteq B_2 \supseteq \cdots

then

\bigcap_^ B_i \in M.

Monotone class theorem for sets

Monotone class theorem for functions

Proof

The following argument originates in

Rick Durrett Richard Timothy Durrett is an American mathematician known for his research and books on mathematical probability theory, stochastic processes and their application to mathematical ecology and population genetics. Education and career He rece ...

's Probability: Theory and Examples.

Results and applications

As a corollary, if

G

is a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

of sets, then the smallest monotone class containing it coincides with the 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. 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

References

* {{Families of sets Families of sets Theorems in measure theory fr:Lemme de classe monotone