alternative definition of Lebesgue integral, an

The standard way of defining Lebesgue integralMathworldPlanetmath is first to define it for simple functionsMathworldPlanetmath, and then to take limits for arbitrary positive measurable functionsMathworldPlanetmath.

There is also another way which uses the Riemann integral [1].

Let (X,,μ) be a measure spaceMathworldPlanetmath. Let f:x+{0} be a nonnegative measurable function. We will define f𝑑μ in [0,] and will call it as the Lebesgue integral of f.

If there exists a t>0 such that μ({x:f(x)>t})=, then we define f𝑑μ=.

Otherwise, assume μ({x:f(x)>t})< for all t(0,) and let Ff(t)=μ({x:f(x)>t}). Ff(t) is a monotonically non-increasing function on (0,), therefore its Riemann integral is well defined on any interval [a,b](0,), so it exists as an improper Riemann integral on (0,). We define


The definition can be extended first to real-valued functions, then complex valued functions as usual.


  • 1 Lieb, E. H., Loss, M., AnalysisMathworldPlanetmath, AMS, 2001.
Title alternative definition of Lebesgue integral, an
Canonical name AlternativeDefinitionOfLebesgueIntegralAn
Date of creation 2013-03-22 17:32:46
Last modified on 2013-03-22 17:32:46
Owner Gorkem (3644)
Last modified by Gorkem (3644)
Numerical id 6
Author Gorkem (3644)
Entry type Definition
Classification msc 26A42
Classification msc 28A25