functional monotone class theorem

The monotone class theorem is a result in measure theory which allows statements about particularly simple classes of functions to be generalized to arbitrary measurable and bounded functions.

Theorem 1.

Let (X,A) be a measurable spaceMathworldPlanetmathPlanetmath and S be a π-system ( generating the σ-algebraMathworldPlanetmathPlanetmath ( A. Suppose that H be a vector spaceMathworldPlanetmath of real-valued functions on X containing the constant functions and satisfying the following,

Then, H contains every bounded and measurable functionMathworldPlanetmath from X to R.

That is a vector space just means that it is closed underPlanetmathPlanetmath taking linear combinationsMathworldPlanetmath, so λf+μg whenever f,g and λ,μ.

As an example application, consider Fubini’s theoremMathworldPlanetmath (, which states that for any two finite measure spaces (X,𝒜,μ) and (Y,,ν) then we may commute the order of integration

f(x,y)𝑑μ(x)𝑑ν(y)=f(x,y)𝑑ν(y)𝑑μ(x). (1)

Here, f:X×Y is a bounded and 𝒜-measurable function. The space of functions for which this identityPlanetmathPlanetmathPlanetmath holds is easily shown to be linearly closed and, by the monotone convergence theoremMathworldPlanetmath, is closed under taking monotone limits of functions. Furthermore, the π-system of sets of the form A×B for A𝒜, B generates the σ-algebra 𝒜 and


So, 1A×B and the monotone class theorem allows us to conclude that all real valued and bounded 𝒜-measurable functions are in , and equation (1) is satisfied.

An alternative, more general form of the monotone class theorem is as follows. The version of the monotone class theorem given above follows from this by letting 𝒦 be the characteristic functions 1A for A𝒮.

Theorem 2.

Let X be a set and K be a collectionMathworldPlanetmath of bounded and real valued functions on X which is closed under multiplicationPlanetmathPlanetmath, so that fgK for all f,gK. Let A be the σ-algebra on X generated by K.

Suppose that H is a vector space of real valued functions on X containing K and the constant functions, and satisfying the following

  • if f:X is bounded and there is a sequence of nonnegative functions fn increasing pointwise to f, then f.

Then, H contains every bounded and real valued A-measurable function on X.

Saying that 𝒜 is the σ-algebra generated by 𝒦 means that it is the smallest σ-algebra containing the sets f-1(B) for f𝒦 and Borel subset B of .

For example, letting 𝒦 be as in theorem 2 and μ,ν be finite measuresMathworldPlanetmath on (X,𝒜) such that μ(X)=ν(X) and f𝑑μ=f𝑑ν for all f𝒦, then f𝑑μ=f𝑑ν for all bounded and measurable real valued functions f. Therefore, μ=ν. In particular, a finite measure μ on (,()) is uniquely determined by its characteristic function χ(x)eixy𝑑μ(y) for x. Similarly, a finite measure μ on a bounded interval is uniquely determined by the integrals xn𝑑μ(x) for n+.

Title functional monotone class theorem
Canonical name FunctionalMonotoneClassTheorem
Date of creation 2013-03-22 18:38:36
Last modified on 2013-03-22 18:38:36
Owner gel (22282)
Last modified by gel (22282)
Numerical id 4
Author gel (22282)
Entry type Theorem
Classification msc 28A20
Related topic MonotoneClassTheorem