properties of the Lebesgue integral of nonnegative measurable functions


Theorem.

Let (X,B,μ) be a measure spaceMathworldPlanetmath, f:X[0,] and g:X[0,] be measurable functionsMathworldPlanetmath, and A,BB. Then the following properties hold:

  1. 1.

    Af𝑑μ0

  2. 2.

    If fg, then Af𝑑μAg𝑑μ.

  3. 3.

    Af𝑑μ=XχAf𝑑μ, where χA denotes the characteristic functionMathworldPlanetmathPlanetmathPlanetmathPlanetmath of A

  4. 4.

    If AB, then Af𝑑μBf𝑑μ.

  5. 5.

    If c0, then Acf𝑑μ=cAf𝑑μ.

  6. 6.

    If μ(A)=0, then Af𝑑μ=0.

  7. 7.

    A(f+g)𝑑μ=Af𝑑μ+Ag𝑑μ

  8. 8.

    If AB=, then ABf𝑑μ=Af𝑑μ+Bf𝑑μ.

  9. 9.

    If f=g almost everywhere with respect to μ, then Af𝑑μ=Ag𝑑μ.

Proof.
  1. 1.

    Let s be a simple functionMathworldPlanetmathPlanetmath with 0sf. Let s=k=1nckχAk for ck[0,] and Ak𝔅. Then As𝑑μ=k=1nckμ(AAk)0. By definition, Af𝑑μAs𝑑μ. It follows that Af𝑑μ0.

  2. 2.

    Let s be a simple function with 0sf. Since fg, 0sg. By definition, As𝑑μAg𝑑μ. Since this holds for every simple function s with 0sf, it follows that Af𝑑μAg𝑑μ.

  3. 3.

    Let s be a simple function with 0sf. Then 0χAsχAf. Let s=k=1nckχAk for ck[0,] and Ak𝔅. Then

    As𝑑μ=k=1nckμ(AAk)=Xk=1nckχAAkdμ=Xk=1nckχAχAkdμ=XχAk=1nckχAkdμ=XχAs𝑑μXχAf𝑑μ.

    Thus, Af𝑑μXχAf𝑑μ.

    Let t be a simple function with 0tχAf. Then χAt=t. Thus, Xt𝑑μ=XχAt𝑑μ=At𝑑μ. Therefore, XχAf𝑑μ=AχAf𝑑μ. Since χAff, AχAf𝑑μAf𝑑μ by property 2. Hence, Af𝑑μXχAf𝑑μ=AχAf𝑑μAf𝑑μ. It follows that Af𝑑μ=XχAf𝑑μ.

  4. 4.

    Since AB, χAχB. Thus, χAfχBf. By property 2, XχAf𝑑μXχBf𝑑μ. By property 3, Af𝑑μ=XχAf𝑑μXχBf𝑑μ=Bf𝑑μ.

  5. 5.

    If c=0, then Acf𝑑μ=A0𝑑μ=0=0Af𝑑μ=cAf𝑑μ.

    If c>0, let S={s:X[0,]sis simple and scf} and

    T={t:X[0,]tis simple and tf}. Then Acf𝑑μ=supsSAs𝑑μ=supsSAcsc𝑑μ=csupsSAsc𝑑μ=csuptTAt𝑑μ=cAf𝑑μ.

  6. 6.

    Let s be a simple function with 0sf. Let s=k=1nckχAk for ck[0,] and Ak𝔅. Then As𝑑μ=k=1nckμ(AAk)=k=1nck0=0. Thus, Af𝑑μ=0.

  7. 7.

    Let {sn} be a nondecreasing sequence of nonnegative simple functions converging pointwise to f and {tn} be a nondecreasing sequence of nonnegative simple functions converging pointwise to g. Then {sn+tn} is a nondecreasing sequence of nonnegative simple functions converging pointwise to f+g. Note that, for every n, A(sn+tn)𝑑μ=Asn𝑑μ+Atn𝑑μ. By Lebesgue’s monotone convergence theorem, A(f+g)𝑑μ=Af𝑑μ+Ag𝑑μ.

  8. 8.

    ABf𝑑μ=XχABf𝑑μ=X(χA+χB-χAB)f𝑑μ=X(χA+χB-χ)f𝑑μ=X(χA+χB-0)f𝑑μ=X(χAf+χBf)𝑑μ=XχAf𝑑μ+XχBf𝑑μ=Af𝑑μ+Bf𝑑μ

  9. 9.

    Let E={xA:f(x)=g(x)}. Since f and g are measurable functions and A𝔅, it must be the case that E𝔅. Thus, AE𝔅. By hypothesisMathworldPlanetmath, μ(AE)=0. Note that E(AE)= and E(AE)=A. Thus, Af𝑑μ=Ef𝑑μ+AEf𝑑μ=Ef𝑑μ+0=Eg𝑑μ+0=Eg𝑑μ+AEg𝑑μ=Ag𝑑μ.

Title properties of the Lebesgue integral of nonnegative measurable functions
Canonical name PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions
Date of creation 2013-03-22 16:13:50
Last modified on 2013-03-22 16:13:50
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 22
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 26A42
Classification msc 28A25
Related topic PropertiesOfTheLebesgueIntegralOfLebesgueIntegrableFunctions