Integral of Integrable Function is Homogeneous
Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f: X \to \overline \R$ be a $\mu$-integrable function.
Let $\lambda \in \R$.
Let $\lambda f$ be the pointwise $\lambda$-multiple of $f$.
Then $\lambda f$ is $\mu$-integrable, and:
- $\ds \int \lambda f \rd \mu = \lambda \int f \rd \mu$
Proof
First suppose that $\lambda \ge 0$.
From Positive Part of Multiple of Function, we have:
- $\paren {\lambda f}^+ = \lambda f^+$
From Negative Part of Multiple of Function, we have:
- $\paren {\lambda f}^- = \lambda f^-$
From Function Measurable iff Positive and Negative Parts Measurable, we have:
- $f^-$ and $f^+$ are $\Sigma$-measurable.
From Pointwise Scalar Multiple of Measurable Function is Measurable, we have:
- $\lambda f^-$, $\lambda f^+$ and $\lambda f$ are $\Sigma$-measurable.
Then, we have:
\(\ds \int \paren {\lambda f}^+ \rd \mu\) | \(=\) | \(\ds \int \lambda f^+ \rd \mu\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \int f^+ \rd \mu\) | Integral of Positive Measurable Function is Positive Homogeneous | |||||||||||
\(\ds \) | \(<\) | \(\ds \infty\) | since $f$ is $\mu$-integrable |
Similarly:
\(\ds \int \paren {\lambda f}^- \rd \mu\) | \(=\) | \(\ds \int \lambda f^- \rd \mu\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \int f^- \rd \mu\) | Integral of Positive Measurable Function is Positive Homogeneous | |||||||||||
\(\ds \) | \(<\) | \(\ds \infty\) | since $f$ is $\mu$-integrable |
So:
- $\lambda f$ is $\mu$-integrable.
We then have:
\(\ds \int \lambda f \rd \mu\) | \(=\) | \(\ds \int \paren {\lambda f}^+ \rd \mu - \int \paren {\lambda f}^- \rd \mu\) | Definition of Integral of Integrable Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \int f^+ \rd \mu - \lambda \int f^- \rd \mu\) | Positive Part of Multiple of Function, Negative Part of Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \paren {\int f^+ \rd \mu - \int f^- \rd \mu}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \int f \rd \mu\) | Definition of Integral of Integrable Function |
Now suppose that $\lambda < 0$.
From Positive Part of Multiple of Function, we have:
- $\paren {\lambda f}^+ = -\lambda f^-$
From Negative Part of Multiple of Function, we have:
- $\paren {\lambda f}^- = -\lambda f^+$
From Function Measurable iff Positive and Negative Parts Measurable, we have:
- $f^-$ and $f^+$ are $\Sigma$-measurable.
From Pointwise Scalar Multiple of Measurable Function is Measurable, we have:
- $\lambda f^-$, $\lambda f^+$ and $\lambda f$ are $\Sigma$-measurable.
Then we have:
\(\ds \int \paren {\lambda f}^+ \rd \mu\) | \(=\) | \(\ds \int \paren {-\lambda f^-} \rd \mu\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\lambda \int f^- \rd \mu\) | Integral of Positive Measurable Function is Positive Homogeneous, since $-\lambda \ge 0$ | |||||||||||
\(\ds \) | \(<\) | \(\ds \infty\) | since $f$ is $\mu$-integrable |
Similarly:
\(\ds \int \paren {\lambda f}^- \rd \mu\) | \(=\) | \(\ds \int \paren {-\lambda f^+} \rd \mu\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\lambda \int f^+ \rd \mu\) | Integral of Positive Measurable Function is Positive Homogeneous | |||||||||||
\(\ds \) | \(<\) | \(\ds \infty\) | since $f$ is $\mu$-integrable |
So:
- $\lambda f$ is $\mu$-integrable.
We then have:
\(\ds \int \lambda f \rd \mu\) | \(=\) | \(\ds \int \paren {\lambda f}^+ \rd \mu - \int \paren {\lambda f}^- \rd \mu\) | Definition of Integral of Integrable Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \paren {-\lambda f^-} \rd \mu - \int \paren {-\lambda f^+} \rd \mu\) | Positive Part of Multiple of Function, Negative Part of Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \int f^+ \rd \mu - \lambda \int f^- \rd \mu\) | Integral of Positive Measurable Function is Positive Homogeneous | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \paren {\int f^+ \rd \mu - \int f^- \rd \mu}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \int f \rd \mu\) | Definition of Integral of Integrable Function |
$\blacksquare$
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $10.4 \ \text{(i)}$