Integral with respect to Pushforward Measure/Corollary
Corollary
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $\struct {X', \Sigma'}$ be a measurable space.
Let $T: X \to X'$ be a $\Sigma \, / \, \Sigma'$-measurable mapping.
Let $f: X' \to \overline \R$ be a positive $\Sigma'$-measurable function.
Let $\map T \mu$ be the pushforward measure of $\mu$ under $T$.
Then $f \circ T : X \to \overline \R$ is $\mu$-integrable if and only if $f : X' \to \overline \R$ is $\map T \mu$-integrable.
In this case, we have:
- $\ds \int_{X'} f \rd \map T \mu = \int_X f \circ T \rd \mu$
Proof
From Function Measurable iff Positive and Negative Parts Measurable, we have:
- $f^+$ and $f^-$ are $\Sigma'$-measurable.
From Composition of Measurable Mappings is Measurable, we have:
- $f \circ T$ is $\Sigma$-measurable.
Then from Function Measurable iff Positive and Negative Parts Measurable, we have:
- $\paren {f \circ T}^+$ and $\paren {f \circ T}^-$ are $\Sigma$-measurable.
Then, we have:
\(\ds \int_X \paren {f \circ T}^+ \rd \mu\) | \(=\) | \(\ds \int_X f^+ \circ T \rd \mu\) | Positive Part of Composition of Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_{X'} f^+ \rd \map T \mu\) | Integral with respect to Pushforward Measure |
and:
\(\ds \int_X \paren {f \circ T}^- \rd \mu\) | \(=\) | \(\ds \int_X f^- \circ T \rd \mu\) | Negative Part of Composition of Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_{X'} f^- \rd \map T \mu\) | Integral with respect to Pushforward Measure |
So:
- $\ds \int_X \paren {f \circ T}^+ \rd \mu < \infty$ and $\ds \int_X \paren {f \circ T}^- \rd \mu <\infty$
- $\ds \int_{X'} f^+ \rd \map T \mu < \infty$ and $\ds \int_{X'} f^- \rd \map T \mu < \infty$.
So $f \circ T : X \to \overline \R$ is $\mu$-integrable if and only if $f : X' \to \overline \R$ is $\map T \mu$-integrable
In this case:
\(\ds \int_{X'} f \rd \map T \mu\) | \(=\) | \(\ds \int_{X'} f^+ \rd \map T \mu - \int_{X'} f^- \rd \map T \mu\) | Definition of Integral of Integrable Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_X \paren {f \circ T}^+ \rd \mu - \int_X \paren {f \circ T}^- \rd \mu\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_X f \circ T \rd \mu\) | Definition of Integral of Integrable Function |
$\blacksquare$
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $14.2$