monotone convergence theorem

Let X be a measure spaceMathworldPlanetmath, and let 0f1f2 be a monotone increasing sequenceMathworldPlanetmath of nonnegative measurable functionsMathworldPlanetmath. Let f:X{} be the function defined by f(x)=limnfn(x). Then f is measurable, and


Remark. This theorem is the first of several theorems which allow us to “exchange integration and limits”. It requires the use of the Lebesgue integralMathworldPlanetmath: with the Riemann integral, we cannot even formulate the theorem, lacking, as we do, the conceptMathworldPlanetmath of “almost everywhere”. For instance, the characteristic functionMathworldPlanetmathPlanetmathPlanetmathPlanetmath of the rational numbers in [0,1] is not Riemann integrable, despite being the limit of an increasing sequence of Riemann integrable functions.

Title monotone convergence theoremMathworldPlanetmath
Canonical name MonotoneConvergenceTheorem
Date of creation 2013-03-22 12:47:27
Last modified on 2013-03-22 12:47:27
Owner Koro (127)
Last modified by Koro (127)
Numerical id 9
Author Koro (127)
Entry type Theorem
Classification msc 26A42
Classification msc 28A20
Synonym Lebesgue’s monotone convergence theorem
Synonym Beppo Levi’s theorem
Related topic DominatedConvergenceTheorem
Related topic FatousLemma