I am taking a course in measure theory and analysis, and just covered this famous theorem which I do not have a name for, therefore I will use the its statement.
Let $U,V\subset R^n$ be open sets. Let $\phi:U \rightarrow V$ be a $C^1$ diffeomorphism between these sets. For any Lebesgue measurable function $f:U \rightarrow C, f\circ\phi:U \rightarrow C$ is Lebesgue measurable, and satisfies
$\int_V f dm = \int_U (f\circ \phi) |J_\phi|dm$,
where $J_\phi(x) = det(Df(x))$, Jacobian determinant. Please ignore integrability assumption omitted in the statement. My question is whether this theorem is useful in Prob Theory/Statistics (some adaptation of it for probability measures, of course)? I was trying to think about nonlinear transformations of random vectors, but I do not see how it can fit there.
This is very useful in probability theory. Suppose $X$ is an $\mathbb{R}^n$-valued random variable that has a density function $f_X : U \to [0, \infty)$ where $U \subset \mathbb{R}^n$ is open. Now suppose you have the random variable $Y = \phi(X)$, where $\phi : U \to V \subset \mathbb{R}^n$ is, say, a diffeomorphism onto the open set $V \subset \mathbb{R}^n$. Then you may wonder whether $Y$ has a density $f_Y$ as well and what the density function is in terms of $f_X$. To find $f_Y$ we compute that for $A \subset V$, \begin{align} P(Y \in A) &= P(X \in \phi^{-1}(A)) \\ &= \int_{\phi^{-1}(A)}f_X(x)\,dx. \end{align} Now make the substitution $y = \phi(x)$, $dy = |J_{\phi}(x)|\,dx$ using your theorem to get $$P(Y \in A) = \int_{\phi^{-1}(A)}f_X(x)\,dx = \int_{A}f_X(x)|J_{\phi}(x)|^{-1}\,dy.$$ This means exactly that $$f_Y(y) = f_X(x)|J_{\phi}(x)|^{-1}.$$ If $\phi$ is not a diffeomorphism but there are open sets $U_1, \dots, U_N$ where $\phi : U_i \to V$ is a diffeomorphism for each $i$, e.g. $\phi(x) = x^2$ with $U_1 = (-\infty, 0)$, $U_2 = (0, \infty)$, you can do a similar comutation and apply your theorem on pieces where $\phi$ is a diffeomorphism to get $$f_Y(y) = \sum_{i = 1}^{N}f_X(x_i)|J_{\phi}(x_i)|^{-1},$$ where $x_i = \phi|_{U_i}^{-1}(y)$.