properties of the Lebesgue integral of Lebesgue integrable functions
Theorem.
Let (X,B,μ) be a measure space, f:X→[-∞,∞] and g:X→[-∞,∞] be Lebesgue integrable
functions, and A,B∈B. Then the following properties hold:
-
1.
|∫Af𝑑μ|≤∫A|f|𝑑μ
-
2.
If f≤g, then ∫Af𝑑μ≤∫Ag𝑑μ.
-
3.
∫Af𝑑μ=∫XχAf𝑑μ, where χA denotes the characteristic function
of A
-
4.
If c∈ℝ, then ∫Acf𝑑μ=c∫Af𝑑μ.
-
5.
If μ(A)=0, then ∫Af𝑑μ=0.
-
6.
∫A(f+g)𝑑μ=∫Af𝑑μ+∫Ag𝑑μ.
-
7.
If A∩B=∅, then ∫A∪Bf𝑑μ=∫Af𝑑μ+∫Bf𝑑μ.
-
8.
If f=g almost everywhere with respect to μ, then ∫Af𝑑μ=∫Ag𝑑μ.
Proof.
-
1.
|∫Af𝑑μ| =|∫Af+𝑑μ-∫Af-𝑑μ| by definition ≤|∫Af+𝑑μ|+|∫Af-𝑑μ| by the triangle inequality =∫Af+𝑑μ+∫Af-𝑑μ by the properties of the Lebesgue integral of nonnegative measurable functions (property 1),
=∫A(f++f-)𝑑μ by the properties of the Lebesgue integral of nonnegative measurable functions (http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions) (property 7), =∫A|f|𝑑μ -
2.
Since f≤g, the following must hold:
-
–
f+=max{0,f}≤max{0,g}=g+;
-
–
-f≥-g;
-
–
f-=max{0,-f}≥max{0,-g}=g-.
Thus, by the properties of the Lebesgue integral of nonnegative measurable functions (http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions) (property 2), ∫Af+𝑑μ≤∫Ag+𝑑μ and ∫Af-𝑑μ≥∫Ag-𝑑μ. Therefore, -∫Af-𝑑μ≤-∫Ag-𝑑μ. Hence, ∫Af+𝑑μ-∫Af-𝑑μ≤∫Ag+𝑑μ-∫Af-𝑑μ≤∫Ag+𝑑μ-∫Ag-𝑑μ. It follows that ∫Af𝑑μ≤∫Ag𝑑μ.
-
–
-
3.
∫Af𝑑μ =∫Af+𝑑μ-∫Af-𝑑μ by definition =∫XχAf+𝑑μ-∫XχAf-𝑑μ by the properties of the Lebesgue integral of nonnegative measurable functions (http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions) (property 3), =∫X(χAf)+𝑑μ-∫X(χAf)-𝑑μ =∫XχAf𝑑μ by definition -
4.
If c≥0, then
∫Acf𝑑μ =∫A(cf)+𝑑μ-∫A(cf)-𝑑μ by definition =∫Acf+𝑑μ-∫Acf-𝑑μ =c∫Af+𝑑μ-c∫Af-𝑑μ by the properties of the Lebesgue integral of nonnegative measurable functions (http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions) (property 5) =c(∫Af+𝑑μ-∫Af-𝑑μ) =c∫Af𝑑μ by definition. If c<0, then
∫Acf𝑑μ =∫A(cf)+𝑑μ-∫A(cf)-𝑑μ by definition =∫A(-c)f-𝑑μ-∫A(-c)f+𝑑μ =-c∫Af-𝑑μ+c∫Af+𝑑μ by the properties of the Lebesgue integral of nonnegative measurable functions (http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions) (property 5) =c(-∫Af-𝑑μ+∫Af+𝑑μ) =c∫Af𝑑μ by definition. -
5.
Note that ∫Af+𝑑μ=0 and ∫Af-𝑑μ=0 by the properties of the Lebesgue integral of nonnegative measurable functions (http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions) (property 6). It follows that ∫Af𝑑μ=0.
-
6.
Let {sn} be a nondecreasing sequence
of nonnegative simple functions
converging pointwise to f++g+ and {tn} be a nondecreasing sequence of nonnegative simple functions converging pointwise to f-+g-. Note that, for every n, ∫Asn𝑑μ-∫Atn𝑑μ=∫A(sn-tn)𝑑μ.
Since f and g are integrable and |f+g|≤|f|+|g|, f+g is integrable. Thus,
∫Af𝑑μ+∫Ag𝑑μ =∫Af+𝑑μ-∫Af-𝑑μ+∫Ag+𝑑μ-∫Ag-𝑑μ by definition =∫Af+𝑑μ+∫Ag+𝑑μ-(∫Af-𝑑μ+∫Ag-𝑑μ) =∫A(f++g+)𝑑μ-(∫A(f-+g-)𝑑μ) by the properties of the Lebesgue integral of nonnegative measurable functions (http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions) (property 7) =lim by Lebesgue’s monotone convergence theorem by Lebesgue’s dominated convergence theorem by definition. -
7.
by definition by the properties of the Lebesgue integral of nonnegative measurable functions (http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions) (property 8), by definition -
8.
Let . Since and are measurable functions and , it must be the case that . Thus, . By hypothesis
, . Note that and . Thus,
∎
Title | properties of the Lebesgue integral of Lebesgue integrable functions |
---|---|
Canonical name | PropertiesOfTheLebesgueIntegralOfLebesgueIntegrableFunctions |
Date of creation | 2013-03-22 16:14:01 |
Last modified on | 2013-03-22 16:14:01 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 19 |
Author | Wkbj79 (1863) |
Entry type | Theorem |
Classification | msc 26A42 |
Classification | msc 28A25 |
Related topic | PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions |