----- > [!proposition] Proposition. ([[change of variables under a linear transformation]]) > With $\lambda$ denoting the [[Lebesgue measure]] on $\mathbb{R}^{n}$ and $E \subset \mathbb{R}^{n}$ [[σ-algebra|measurable]], one has [[Lp-norm|for any]] $h \in L^{1}(\lambda)$ [[general linear group|and]] $T \in \operatorname{GL}(\mathbb{R}^{n})$ that $\int_{E} h \circ T \, d \lambda = \frac{1}{|\det T|} \int_{T(E)} h \, d \lambda .$ ^proposition > [!proof]+ Proof. ([[change of variables under a linear transformation]]) > This is a corollary of [[scaling of Lebesgue measure under a linear transformation]]. > > First assume $h=\chi_{E}$ is a [[characteristic function]] for some [[σ-algebra|measurable subset]] $E$. Noting $\chi_{E} \circ T= \chi_{T ^{-1} (E)}$, we have in this case $\begin{align} > \int h \circ T \ \, d \lambda&= \int \chi_{T ^{-1} (E)} \, d \lambda \\ > &= \lambda (T ^{-1} (E)) \\ > &= |\det T ^{-1}| \lambda (E) \\ > &= \frac{\lambda(E)}{|\det T|} \\ > &= \frac{1}{|\det T|} \int h \, d \lambda > \end{align}$ > and the result holds. Next assume $h=s$ is a [[simple function]], $s=\sum_{i=1}^{\ell}a_{i} \chi_{E_{i}}$. Then $\begin{align} > \int h \circ T \, d \lambda &= \int \sum_{i=1}^{\ell} a_{i} \chi_{E_{i}} \circ T \, d \lambda \\ > &= \sum_{i=1}^{\ell} a_{i} |\det T| ^{-1} \lambda(E_{i}) \\ > &= \frac{1}{| \det T|} \sum_{i=1}^{\ell} a_{i} \lambda(E_{i}) \\ > &= \frac{1}{|\det T|} \int h \, d \lambda > \end{align}$ > and the result holds. Finally assume $h \in L^{1}$ arbitrary. By splitting $h=h^{+}-h^{-}$ we may assume $h \geq 0$. The construction in [[approximation by simple functions]] then gives a [[monotonic map|monotone increasing]] [[sequence]] of nonnegative [[simple function|simple functions]] $s_{n}$ [[converge|converging]] [[pointwise convergence|pointwise]] to $h$, $s_{n} \uparrow h$. Thus the [[monotone convergence theorem for nonnegative measurable functions|monotone convergence theorem]] applies to give $\lim_{n \to \infty} \int s_{n} \, d \lambda = \int h \, d \lambda$. Also, $s_{n} \circ T \uparrow h \circ T$, whence $\lim_{n \to \infty} \int (s_{n} \circ T) \ d \lambda=\int h \circ T \, d \lambda$, where we note $\lim_{n \to \infty}(s_{n} \circ T)=h \circ T$ per general considerations regarding pointwise convergence. Putting things together, we obtain $\begin{align} > \int h \circ T \, d \lambda &= \lim_{n \to \infty} \int (s_{n} \circ T) \, d \lambda \\ > &= \lim_{n \to \infty} \int \frac{1}{|\det T|} s_{n} \, d \lambda \\ > &= \frac{1}{|\det T|} \int h \, d \lambda. > \end{align}$ > ^proof ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```