# 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. 1.

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

2. 2.

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

3. 3.

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

4. 4.

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

5. 5.

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

6. 6.

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

7. 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. 8.

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

###### Proof.
1. 1.
$\displaystyle\left|\int_{A}f\,d\mu\right|$ $\displaystyle=\left|\int_{A}f^{+}\,d\mu-\int_{A}f^{-}\,d\mu\right|$ 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. 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 $\displaystyle\int_{A}f\,d\mu\leq\int_{A}g\,d\mu$.

3. 3.
$\displaystyle\int_{A}f\,d\mu$ $\displaystyle=\int_{A}f^{+}\,d\mu-\int_{A}f^{-}\,d\mu$ 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
4. 4.

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. 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 $\displaystyle\int_{A}f\,d\mu=0$.

6. 6.

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.
7. 7.
$\displaystyle\int_{A\cup B}f\,d\mu$ $\displaystyle=\int_{A\cup B}f^{+}\,d\mu-\int_{A\cup B}f^{-}\,d\mu$ 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. 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 PropertiesOfTheLebesgueIntegralOfLebesgueIntegrableFunctions 2013-03-22 16:14:01 2013-03-22 16:14:01 Wkbj79 (1863) Wkbj79 (1863) 19 Wkbj79 (1863) Theorem msc 26A42 msc 28A25 PropertiesOfTheLebesgueIntegralOfNonnegativeMeasurableFunctions