Suppose T is a $1-1$ ($\mathcal{C'}$ mapping of an open set $E \subset R^k$ into $R^k$ such that $J_{T(x)} \ne 0$ for all $x \in E$. If $f$ is a continuous function on $R^k$ whose support is compact and lies in $T(E)$, then
$\int_{R^k}f(y)dy = \int_{R^k}f(T(x))|J_T(x)|dx \cdots (1)$
It follows from the remarks just made that $(1)$ is true if $T$ is a primitive $\mathcal{C'}$-mapping (see Definition 10.5), and Theorem 10.2(which is integral remains the same no matter in which order we integrate the functions ) shows that (1) is true if $T$ is a linear mapping which merely interchanges two coordinates.(What are we trying to prove using this ? How are we proving this?)
If the theorem is true for transformations $P,Q$ and if $S(x) = P(Q(x))$(are we assuming P and Q to be both injective and surjective ? If so then why?) then
$\int f(z)dz = \int f(P(y))|J_P(y)|dy = \int f(P(Q(x))|J_P(Q(x))||J_Q(x)|dx = \int f(S(x))S(x)dx$
Since ,
$J_P(Q(x))J_Q(x) = det P'(Q(x))deet(Q'(x)) = detP'(Q(x))Q'(x) = det (S'(x)) = J_S(x)$ by the multiplication theorem for determinants and chain rule. Thus the theorem is also true for $S$.
Each point $a \in E$ has a neighborhood $U \subset E$ in which
$T(x) = T(a) + B_1...B_{k-1}G_k....G_1(x-a)$(How are we writing this?)
where $G_i$ and $B_i$ are primitive $\mathbb{C'}$ mapping and each $B_i$ is a flip.
Setting $V = T(U)$ it follows that $(1)$ holds if the support of $f$ lies in $V$.(I don't understand this part)
Each point $y \in T(E)$ lies in an open set $V_Y \subset T(E)$ ) such that $(1)$ holds for all continuous functions whose support lies in $V_y$.
Now let $f$ be a continuous function with compact support $K \subset T(E)$.
Since { $V_y$} covers $K$, the $f = \varphi_if$ if, where each $\varphi_i$ is continuous, and each $\varphi_i$ has its support in some $V_y$.
Thus $(1)$ holds for each $\varphi_if$, and hence also for their sum $f$. (I really can't conclude this statement)
Will it be possible for someone to explain the proof in the case when $k =2 $?