Pushforward of Lebesgue Measure under General Linear Group
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Theorem
Let $M \in \GL {n, \R}$ be a nonsingular matrix.
Let $\lambda^n$ be $n$-dimensional Lebesgue measure.
Then the pushforward measure $M_* \lambda^n$ satisfies:
- $M_* \lambda^n = \size {\det M^{-1} } \cdot \lambda^n$
Proof
From Linear Transformation on Euclidean Space is Continuous, $M^{-1}$ is a continuous mapping.
Thus from Continuous Mapping is Measurable, it is measurable, and so $M_* \lambda^n$ is defined.
Now let $B \in \map \BB {\R^n}$ be a Borel measurable set, and let $\mathbf x \in \R^n$.
Then:
\(\ds \map {M_* \lambda^n} {\mathbf x + B}\) | \(=\) | \(\ds \map {\lambda^n} {\map {M^{-1} } {\mathbf x + B} }\) | Definition of Pushforward Measure | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\lambda^n} {\map {M^{-1} } {\mathbf x} + \map {M^{-1} } B}\) | $M^{-1}$ is linear | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\lambda^n} {\map {M^{-1} } B}\) | Lebesgue Measure is Translation Invariant | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {M_* \lambda^n} B\) | Definition of Pushforward Measure |
Thus $M_* \lambda^n$ is a translation invariant measure.
From Translation Invariant Measure on Euclidean Space is Multiple of Lebesgue Measure, it follows that:
- $M_* \lambda^n = \map {M_* \lambda^n} {\openint 0 1^n} \cdot \lambda^n$
Lastly, using Determinant as Volume of Parallelotope it follows that:
- $\map {M_* \lambda^n} {\openint 0 1^n} = \map {\lambda^n} {\map {M^{-1} } {\openint 0 1^n} } = \size {\det M^{-1} }$
Hence the result.
$\blacksquare$
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $4.9 \ \text{(iii)}$
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $7.10$