[proofplan]
We prove the formula by testing the law of $X$ on an arbitrary Borel set $C \subset B$. The event $\{X \in C\}$ is the same as $\{Z \in T^{-1}(C)\}$, so the density of $Z$ writes its probability as an integral over $T^{-1}(C)$. The multivariable change-of-variables theorem applied to the $C^1$ diffeomorphism $T^{-1}:B\to A$ transforms that integral into an integral over $C$ with the proposed integrand. Since this identity holds for every Borel set $C \subset B$, the proposed function is a density of $X$.
[/proofplan]
[step:Define the candidate density on $B$]
Define the function $q:B\to[0,\infty)$ by
\begin{align*}
q(x)=p_Z(T^{-1}(x))\,|\det JT^{-1}_x|.
\end{align*}
The map $T^{-1}:B\to A$ is continuous, hence Borel measurable. The function $p_Z$ is Borel measurable as a density, and the map $x\mapsto JT^{-1}_x$ is continuous because $T^{-1}$ is $C^1$. Therefore $x\mapsto |\det JT^{-1}_x|$ is Borel measurable, so $q$ is Borel measurable and non-negative.
[/step]
[step:Express probabilities of transformed events using the density of $Z$]
Let $C\subset B$ be a Borel set. Since $T^{-1}:B\to A$ is continuous, the preimage $T^{-1}(C)$ is a Borel subset of $A$. By the definition of $X=T\circ Z$,
\begin{align*}
\{ \omega\in\Omega : X(\omega)\in C\}
=
\{ \omega\in\Omega : Z(\omega)\in T^{-1}(C)\}.
\end{align*}
Using the density formula for $Z$ on the Borel set $T^{-1}(C)\subset A$, we get
\begin{align*}
\mathbb P(X\in C)
=
\mathbb P(Z\in T^{-1}(C))
=
\int_{T^{-1}(C)} p_Z(z)\,d\mathcal L^m(z).
\end{align*}
[guided]
Fix a Borel set $C\subset B$. We want to compute the probability that the transformed random vector $X$ lands in $C$. Because $X=T\circ Z$, this event can be rewritten entirely in terms of $Z$:
\begin{align*}
\{ \omega\in\Omega : X(\omega)\in C\}
=
\{ \omega\in\Omega : T(Z(\omega))\in C\}.
\end{align*}
Since $T:A\to B$ is bijective, the condition $T(Z(\omega))\in C$ is equivalent to $Z(\omega)\in T^{-1}(C)$. Hence
\begin{align*}
\{ \omega\in\Omega : X(\omega)\in C\}
=
\{ \omega\in\Omega : Z(\omega)\in T^{-1}(C)\}.
\end{align*}
We must also check that the right-hand event is measurable. The set $C$ is Borel in $B$, and $T^{-1}:B\to A$ is continuous because it is $C^1$. Therefore $T^{-1}(C)$ is Borel in $A$. Since $Z:(\Omega,\mathcal F)\to(A,\mathcal B(A))$ is measurable, the set $\{\omega\in\Omega:Z(\omega)\in T^{-1}(C)\}$ lies in $\mathcal F$.
Now we use the defining property of the density $p_Z$. For every Borel set $E\subset A$,
\begin{align*}
\mathbb P(Z\in E)=\int_E p_Z(z)\,d\mathcal L^m(z).
\end{align*}
Applying this with $E=T^{-1}(C)$ gives
\begin{align*}
\mathbb P(X\in C)
=
\mathbb P(Z\in T^{-1}(C))
=
\int_{T^{-1}(C)} p_Z(z)\,d\mathcal L^m(z).
\end{align*}
[/guided]
[/step]
[step:Apply change of variables through $T^{-1}$]
Apply the Multivariable Change-of-Variables Theorem (citing a result not yet in the wiki: Multivariable Change-of-Variables Theorem) to the $C^1$ diffeomorphism $T^{-1}:B\to A$ and the non-negative Borel function $p_Z:A\to[0,\infty)$. With the substitution
\begin{align*}
z=T^{-1}(x),
\end{align*}
the [Lebesgue measure](/page/Lebesgue%20Measure) transforms by the Jacobian factor $|\det JT^{-1}_x|$. Therefore
\begin{align*}
\int_{T^{-1}(C)} p_Z(z)\,d\mathcal L^m(z)
=
\int_C p_Z(T^{-1}(x))\,|\det JT^{-1}_x|\,d\mathcal L^m(x).
\end{align*}
By the definition of $q$, this is
\begin{align*}
\mathbb P(X\in C)
=
\int_C q(x)\,d\mathcal L^m(x).
\end{align*}
[/step]
[step:Identify the density of $X$]
The preceding identity holds for every Borel set $C\subset B$:
\begin{align*}
\mathbb P(X\in C)=\int_C q(x)\,d\mathcal L^m(x).
\end{align*}
Thus the law of $X$ is absolutely continuous with respect to $\mathcal L^m|_B$, and $q$ is a density of $X$ on $B$. Substituting the definition of $q$ gives
\begin{align*}
p_X(x)=p_Z(T^{-1}(x))\,|\det JT^{-1}_x|
\end{align*}
for $\mathcal L^m$-a.e. $x\in B$. Under the present hypotheses there are no exceptional points where differentiability or invertibility fails; in applications with exceptional null sets, the same conclusion remains valid whenever the change-of-variables theorem applies after ignoring those null sets.
[/step]
