Monotone class theorem

Top

Home > Library > Miscellaneous > Wikipedia

A monotone class in $R$ is a collection $\mathcal{M}$ of subsets of $R$ which is closed under countable monotone unions and intersections, i.e. if $A_i \in \mathcal{M}$ and $A_1 \subset A_2 \subset \ldots$ then $\cup_{i = 1}^\infty A_i \in \mathcal{M}$ , and similarly for intersections of decreasing sequences of sets.

The Monotone Class Theorem says that the smallest monotone class containing an algebra of sets $\mathcal{G}$ is precisely the smallest σ-algebra containing $\mathcal{G}$ .

As a corollary, if $\mathcal{G}$ is a ring of sets, then the smallest monotone class containing it coincides with the sigma-ring of $\mathcal{G}$ .

This theorem is used as a type of transfinite induction, and is used to prove many Theorems, such as Fubini's theorem in basic measure theory.

A functional version of this theorem can be found at PlanetMath.^[1]

References:

^ http://planetmath.org/encyclopedia/FunctionalMonotoneClassTheorem.html

This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

Help us answer these

What a list class a class b class c drugs?

Why are class rules and class procedures important for class learning?

A class generated from a template class is?

Is Cauchy's theorem a theorem in combinatorics?

Post a question - any question - to the WikiAnswers community:

Copyrights:

Cite

Wikipedia on Answers.com This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article Monotone class theorem. Read

Monotone class theorem