Change of variables on bounded open Jordan sets when both integrands are bounded and Riemann integrable

If $U=\emptyset$, then $V=\emptyset$ and both integrals are $0$. Otherwise choose the compact grid exhaustion $K_j\uparrow U$ from [L1]. For every $j$, [L2] gives $\int_{g(K_j)}f=\int_{K_j}h.$

A bound $M_h$ for $|h|$ gives source error at most $M_h\operatorname{cont}(U\setminus K_j)$, which tends to zero by [L1] and [L3].

The compact set $\overline U$ is nonempty, and [L4] gives a bound $M_D$ for $|\det Dg|$ on it. Apply [L2] to the compact Jordan remainder $\overline U\setminus\operatorname{int}K_j$ with the constant-one function. Its content tends to zero with $\operatorname{cont}(U\setminus K_j)$, so $\operatorname{cont}(V\setminus g(K_j))\le M_D\operatorname{cont}(\overline U\setminus\operatorname{int}K_j)\to0$. A bound for $|f|$ and [L3] make the image error tend to zero as well. Passing to the limit in step 1.1 proves the formula.