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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:
- $\displaystyle \left| \int_A f \, d\mu \right| \le \int_A |f| \, d\mu$
- If $f \le g$ , then $\displaystyle \int_A f \, d\mu \le \int_A g \, d\mu$ .
- $\displaystyle \int_A f \, d\mu =\int_X \chi_A f \, d\mu$ , where $\chi_A$ denotes the characteristic function of $A$
- If $c \in \mathbb{R}$ , then $\displaystyle \int_A cf \, d\mu =c\int_A f \, d\mu$ .
- If $\mu(A)=0$ , then $\displaystyle \int_A f \, d\mu =0$ .
- $\displaystyle \int_A (f+g) \, d\mu =\int_A f \, d\mu +\int_A g \, d\mu$ .
- If $A \cap B=\emptyset$ , then $\displaystyle \int_{A \cup B} f \, d\mu =\int_A f \, d\mu +\int_B f \, d\mu$ .
- 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 $\displaystyle \le \left| \int_A f^+ \, d\mu \right| +\left| \int_A f^- \, d\mu \right|$ 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 (property 7), $\displaystyle =\int_A |f| \, d\mu$ - Since $f \le g$ , the following must hold:
- $f^+=\max\{0,f\}\le\max\{0,g\}=g^+$ ;
- $-f \ge -g$ ;
- $f^-=\max\{0,-f\}\ge\max\{0,-g\}=g^-$ .
Thus, by the properties of the Lebesgue integral of nonnegative measurable functions (property 2), $\displaystyle \int_A f^+ \, d\mu \le \int_A g^+ \, d\mu$ and $\displaystyle \int_A f^- \, d\mu \ge \int_A g^- \, d\mu$ . Therefore, $\displaystyle -\int_A f^- \, d\mu \le -\int_A g^- \, d\mu$ . Hence, $\displaystyle \int_A f^+ \, d\mu -\int_A f^- \, d\mu \le \int_A g^+ \, d\mu -\int_A f^- \, d\mu \le \int_A g^+ \, d\mu -\int_A g^- \, d\mu$ . It follows that $\displaystyle \int_A f \, d\mu \le \int_A g \, d\mu$ .
-
$\displaystyle \int_A f \, d\mu$ $\displaystyle =\int_A f^+ \, d\mu -\int_A f^- \, d\mu$ by definition $\displaystyle =\int_X \chi_Af^+ \, d\mu -\int_X \chi_Af^- \, d\mu$ by the properties of the Lebesgue integral of nonnegative measurable functions (property 3), $\displaystyle =\int_X (\chi_Af)^+ \, d\mu -\int_X (\chi_Af)^- \, d\mu$ $\displaystyle =\int_X \chi_Af \, d\mu$ by definition - If $c \ge 0$ , 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 (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 (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. - 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 (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 $|f+g| \le |f|+|g|$ , $f+g$ 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 (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 (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 - 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, $\displaystyle \int_A f \, d\mu =\int_E f \, d\mu +\int_{A \setminus E} f \, d\mu =\int_E f \, d\mu +0=\int_E g \, d\mu +0=\int_E g \, d\mu +\int_{A \setminus E} g \, d\mu =\int_A g \, d\mu.$
properties of the Lebesgue integral of Lebesgue integrable functions is owned by Warren Buck.
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