Vitali convergence theorem

Let f1,f2, be 𝐋p-integrable functions on some measure spaceMathworldPlanetmath, for 1p<.

The sequence {fn} convergesPlanetmathPlanetmath in 𝐋p to a measurable functionMathworldPlanetmath f if and and only if

  1. i

    the sequence {fn} converges to f in measure;

  2. ii

    the functions {|fn|p} are uniformly integrable; and

  3. iii

    for every ϵ>0, there exists a set E of finite measure, such that Ec|fn|p<ϵ for all n.


This theorem can be used as a replacement for the more well-known dominated convergence theorem, when a dominating cannot be found for the functions fn to be integrated. (If this theorem is known, the dominated convergence theorem can be derived as a special case.)

In a finite measure space, condition (iii) is trivial. In fact, condition (iii) is the tool used to reduce considerations in the general case to the case of a finite measure space.

In probability , the definition of “uniform integrability” is slightly different from its definition in general measure theory; either definition may be used in the statement of this theorem.


  • 1 Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications, second ed. Wiley-Interscience, 1999.
  • 2 Jeffrey S. Rosenthal. A First Look at Rigorous Probability Theory. World Scientific, 2003.
Title Vitali convergence theoremMathworldPlanetmath
Canonical name VitaliConvergenceTheorem
Date of creation 2013-03-22 16:17:10
Last modified on 2013-03-22 16:17:10
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 9
Author stevecheng (10074)
Entry type Theorem
Classification msc 28A20
Synonym uniform-integrability convergence theorem
Related topic ModesOfConvergenceOfSequencesOfMeasurableFunctions
Related topic UniformlyIntegrable
Related topic DominatedConvergenceTheorem