monotone class theorem

Monotone Class theoremFernando Sanz Gamiz


Let F0 an algebra of subsets of Ω. Let M be the smallest monotone class such that F0M and σ(F0) be the sigma algebra generated by F0. Then M=σ(F0).


It is enough to prove that is an algebra, because an algebra which is a monotone class is obviously a σ-algebra.

Let A={B|AB,AB and AB}. Then is clear that A is a monotone class and, in fact, A=, for if A0, then 0A since 0 is a field, hence A by minimality of ; consequently =A by definition of A. But this shows that for any B we have AB,AB and AB for any A0, so that 0B and again by minimality =B. But what we have just proved is that is an algebra, for if A,B=A we have showed that AB,AB and AB, and, of course, Ω. ∎

Remark 1.

One of the main applications of the Monotone Class Theorem is that of showing that certain property is satisfied by all sets in an σ-algebra, generally starting by the fact that the field generating the σ-algebra satisfies such property and that the sets that satisfies it constitutes a monotone class.

Example 1.

Consider an infiniteMathworldPlanetmathPlanetmath sequenceMathworldPlanetmath of independent random variablesMathworldPlanetmath {Xn,n}. The definition of independence is


for any Borel sets A1,A2,..,An and any finite n. Using the Monotone Class Theorem one can show, for example, that any event in σ(X1,X2,,Xn) is independent of any event in σ(Xn+1,Xn+2,). For, by independence


when A and B are measurable rectangles in n and respectively. Now it is clear that the sets A which satisfies the above relationMathworldPlanetmathPlanetmath form a monotone class. So


for every Aσ(X1,X2,,Xn) and any measurable rectangle B. A second application of the theorem shows finally that the above relation holds for any Aσ(X1,X2,,Xn) and Bσ(Xn+1,Xn+2,)

Title monotone class theorem
Canonical name MonotoneClassTheorem
Date of creation 2013-03-22 17:07:34
Last modified on 2013-03-22 17:07:34
Owner fernsanz (8869)
Last modified by fernsanz (8869)
Numerical id 8
Author fernsanz (8869)
Entry type Theorem
Classification msc 28A05
Related topic MonotoneClass
Related topic SigmaAlgebra
Related topic Algebra
Related topic FunctionalMonotoneClassTheorem