properties of the Lebesgue integral of Lebesgue integrable functions

Theorem.

Let $(X,\mathfrak{B},\mu)$ be a measure space, $f\colon X\to[-\infty,\infty]$ and $g\colon X\to[-\infty,\infty]$ be Lebesgue integrable functions, and $A,B\in\mathfrak{B}$ . Then the following properties hold:

1.

$\displaystyle\left|\int_{A}f\,d\mu\right|\leq\int_{A}|f|\,d\mu$
2.

If $f\leq g$ , then $\displaystyle\int_{A}f\,d\mu\leq\int_{A}g\,d\mu$ .
3.

$\displaystyle\int_{A}f\,d\mu=\int_{X}\chi_{A}f\,d\mu$ , where $\chi_{A}$ denotes the characteristic function of $A$
4.

If $c\in\mathbb{R}$ , then $\displaystyle\int_{A}cf\,d\mu=c\int_{A}f\,d\mu$ .
5.

If $\mu(A)=0$ , then $\displaystyle\int_{A}f\,d\mu=0$ .
6.

$\displaystyle\int_{A}(f+g)\,d\mu=\int_{A}f\,d\mu+\int_{A}g\,d\mu$ .
7.

If $A\cap B=\emptyset$ , then $\displaystyle\int_{A\cup B}f\,d\mu=\int_{A}f\,d\mu+\int_{B}f\,d\mu$ .
8.

If $f=g$ almost everywhere with respect to $\mu$ , then $\displaystyle\int_{A}f\,d\mu=\int_{A}g\,d\mu$ .

Proof.

$\displaystyle\left\|\int_{A}f\,d\mu\right\|$	$\displaystyle=\left\|\int_{A}f^{+}\,d\mu-\int_{A}f^{-}\,d\mu\right\|$ by definition
	$\displaystyle\leq\left\|\int_{A}f^{+}\,d\mu\right\|+\left\|\int_{A}f^{-}\,d\mu\right\|$ by the triangle inequality
	$\displaystyle=\int_{A}f^{+}\,d\mu+\int_{A}f^{-}\,d\mu$ by the
	properties of the Lebesgue integral of nonnegative measurable functions (property 1),
	$\displaystyle=\int_{A}(f^{+}+f^{-})\,d\mu$ by the
	properties of the Lebesgue integral of nonnegative measurable functions (http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions) (property 7),
	$\displaystyle=\int_{A}\|f\|\,d\mu$

2.

Since $f\leq g$ , the following must hold:
- –
  
  $f^{+}=\max\{0,f\}\leq\max\{0,g\}=g^{+}$ ;
- –
  
  $-f\geq-g$ ;
- –
  
  $f^{-}=\max\{0,-f\}\geq\max\{0,-g\}=g^{-}$ .
Thus, by the properties of the Lebesgue integral of nonnegative measurable functions (http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions) (property 2), $\displaystyle\int_{A}f^{+}\,d\mu\leq\int_{A}g^{+}\,d\mu$ and $\displaystyle\int_{A}f^{-}\,d\mu\geq\int_{A}g^{-}\,d\mu$ . Therefore, $\displaystyle-\int_{A}f^{-}\,d\mu\leq-\int_{A}g^{-}\,d\mu$ . Hence, $\displaystyle\int_{A}f^{+}\,d\mu-\int_{A}f^{-}\,d\mu\leq\int_{A}g^{+}\,d\mu-% \int_{A}f^{-}\,d\mu\leq\int_{A}g^{+}\,d\mu-\int_{A}g^{-}\,d\mu$ . It follows that $\displaystyle\int_{A}f\,d\mu\leq\int_{A}g\,d\mu$ .

$\displaystyle\int_{A}f\,d\mu$	$\displaystyle=\int_{A}f^{+}\,d\mu-\int_{A}f^{-}\,d\mu$ by definition
	$\displaystyle=\int_{X}\chi_{A}f^{+}\,d\mu-\int_{X}\chi_{A}f^{-}\,d\mu$ by the
	properties of the Lebesgue integral of nonnegative measurable functions (http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions) (property 3),
	$\displaystyle=\int_{X}(\chi_{A}f)^{+}\,d\mu-\int_{X}(\chi_{A}f)^{-}\,d\mu$
	$\displaystyle=\int_{X}\chi_{A}f\,d\mu$ by definition

If $c\geq 0$ , then

$\displaystyle\int_{A}cf\,d\mu$	$\displaystyle=\int_{A}(cf)^{+}\,d\mu-\int_{A}(cf)^{-}\,d\mu$ by definition
	$\displaystyle=\int_{A}cf^{+}\,d\mu-\int_{A}cf^{-}\,d\mu$
	$\displaystyle=c\int_{A}f^{+}\,d\mu-c\int_{A}f^{-}\,d\mu$ by the
	properties of the Lebesgue integral of nonnegative measurable functions (http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions) (property 5)
	$\displaystyle=c\left(\int_{A}f^{+}\,d\mu-\int_{A}f^{-}\,d\mu\right)$
	$\displaystyle=c\int_{A}f\,d\mu$ by definition.

If $c<0$ , then

$\displaystyle\int_{A}cf\,d\mu$	$\displaystyle=\int_{A}(cf)^{+}\,d\mu-\int_{A}(cf)^{-}\,d\mu$ by definition
	$\displaystyle=\int_{A}(-c)f^{-}\,d\mu-\int_{A}(-c)f^{+}\,d\mu$
	$\displaystyle=-c\int_{A}f^{-}\,d\mu+c\int_{A}f^{+}\,d\mu$ by the
	properties of the Lebesgue integral of nonnegative measurable functions (http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions) (property 5)
	$\displaystyle=c\left(-\int_{A}f^{-}\,d\mu+\int_{A}f^{+}\,d\mu\right)$
	$\displaystyle=c\int_{A}f\,d\mu$ by definition.

5.

Note that $\displaystyle\int_{A}f^{+}\,d\mu=0$ and $\displaystyle\int_{A}f^{-}\,d\mu=0$ by the properties of the Lebesgue integral of nonnegative measurable functions (http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions) (property 6). It follows that $\displaystyle\int_{A}f\,d\mu=0$ .

Let $\{s_{n}\}$ be a nondecreasing sequence of nonnegative simple functions converging pointwise to $f^{+}+g^{+}$ and $\{t_{n}\}$ be a nondecreasing sequence of nonnegative simple functions converging pointwise to $f^{-}+g^{-}$ . Note that, for every $n$ , $\displaystyle\int_{A}s_{n}\,d\mu-\int_{A}t_{n}\,d\mu=\int_{A}(s_{n}-t_{n})\,d\mu$ .

Since $f$ and $g$ are integrable and $|f+g|\leq|f|+|g|$ , $f+g$ is integrable. Thus,

$\displaystyle\int_{A}f\,d\mu+\int_{A}g\,d\mu$	$\displaystyle=\int_{A}f^{+}\,d\mu-\int_{A}f^{-}\,d\mu+\int_{A}g^{+}\,d\mu-\int% _{A}g^{-}\,d\mu$ by definition
	$\displaystyle=\int_{A}f^{+}\,d\mu+\int_{A}g^{+}\,d\mu-\left(\int_{A}f^{-}\,d% \mu+\int_{A}g^{-}d\mu\right)$
	$\displaystyle=\int_{A}(f^{+}+g^{+})\,d\mu-\left(\int_{A}(f^{-}+g^{-})\,d\mu\right)$ by the
	properties of the Lebesgue integral of nonnegative measurable functions (http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions) (property 7)
	$\displaystyle=\lim_{n\to\infty}\int_{A}s_{n}\,d\mu-\left(\lim_{n\to\infty}\int% _{A}t_{n}\,d\mu\right)$ by Lebesgue’s monotone convergence theorem
	$\displaystyle=\lim_{n\to\infty}\left(\int_{A}s_{n}\,d\mu-\int_{A}t_{n}\,d\mu\right)$
	$\displaystyle=\lim_{n\to\infty}\int_{A}(s_{n}-t_{n})\,d\mu$
	$\displaystyle=\int_{A}(f^{+}+g^{+}-(f^{-}+g^{-}))\,d\mu$ by Lebesgue’s dominated convergence theorem
	$\displaystyle=\int_{A}(f^{+}-f^{-}+g^{+}-g^{-})\,d\mu$
	$\displaystyle=\int_{A}(f+g)\,d\mu$ by definition.

$\displaystyle\int_{A\cup B}f\,d\mu$	$\displaystyle=\int_{A\cup B}f^{+}\,d\mu-\int_{A\cup B}f^{-}\,d\mu$ by definition
	$\displaystyle=\int_{A}f^{+}\,d\mu+\int_{B}f^{+}\,d\mu-\left(\int_{A}f^{-}\,d% \mu+\int_{B}f^{-}\,d\mu\right)$ by the
	properties of the Lebesgue integral of nonnegative measurable functions (http://planetmath.org/PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions) (property 8),
	$\displaystyle=\int_{A}f^{+}\,d\mu-\int_{A}f^{-}\,d\mu+\int_{B}f^{+}\,d\mu-\int% _{B}f^{-}\,d\mu$
	$\displaystyle=\int_{A}f\,d\mu+\int_{B}f\,d\mu$ by definition

8.

Let $E=\{x\in A:f(x)=g(x)\}$ . Since $f$ and $g$ are measurable functions and $A\in\mathfrak{B}$ , it must be the case that $E\in\mathfrak{B}$ . Thus, $A-E\in\mathfrak{B}$ . By hypothesis, $\mu(A\setminus E)=0$ . Note that $E\cap(A\setminus E)=\emptyset$ and $E\cup(A\setminus E)=A$ . Thus, $\displaystyle\int_{A}f\,d\mu=\int_{E}f\,d\mu+\int_{A\setminus E}f\,d\mu=\int_{% E}f\,d\mu+0=\int_{E}g\,d\mu+0=\int_{E}g\,d\mu+\int_{A\setminus E}g\,d\mu=\int_% {A}g\,d\mu.$

∎

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