proof of dominated convergence theorem

Define the functions hn+ and hn- as follows:


These suprema and infima exist because, for every x, |fn(x)|g(x). These functions enjoy the following properties:

For every n, |hn±|g

The sequenceMathworldPlanetmath hn+ is decreasing and the sequence hn- is increasing.

For every x, limnhn±(x)=f(x)

Each hn± is measurable.

The first property follows from immediately from the definition of supremum. The second property follows from the fact that the supremum or infimumMathworldPlanetmath is being taken over a larger set to define hn±(x) than to define hm±(x) when n>m. The third property is a simple consequence of the fact that, for any sequence of real numbers, if the sequence convergesPlanetmathPlanetmath, then the sequence has an upper limitMathworldPlanetmath and a lower limit which equal each other and equal the limit. As for the fourth statement, it means that, for every real number y and every integer n, the sets

{xhn-(x)y} and {xhn+(x)y}

are measurable. However, by the definition of hn±, these sets can be expressed as

mn{xfn(x)y} and mn{xfn(x)y}

respectively. Since each fn is assumed to be measurable, each set in either union is measurable. Since the union of a countableMathworldPlanetmath number of measurable setsMathworldPlanetmath is itself measurable, these unions are measurable, and hence the functions hn± are measurable.

Because of properties 1 and 4 above and the assumptionPlanetmathPlanetmath that g is integrable, it follows that each hn± is integrable. This conclusionMathworldPlanetmath and property 2 mean that the monotone convergence theoremMathworldPlanetmath is applicable so one can conclude that f is integrable and that


By property 3, the right hand side equals f(x)𝑑μ(x).

By construction, hn-fnhn+ and hence


Because the outer two terms in the above inequalityMathworldPlanetmath tend towards the same limit as n, the middle term is squeezed into converging to the same limit. Hence

Title proof of dominated convergence theoremPlanetmathPlanetmath
Canonical name ProofOfDominatedConvergenceTheorem1
Date of creation 2013-03-22 14:33:58
Last modified on 2013-03-22 14:33:58
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 5
Author rspuzio (6075)
Entry type Proof
Classification msc 28A20
Related topic ProofOfDominatedConvergenceTheorem