 Lebesgue substitution on an interval

Let $\varphi :I\to J$ be a $C^1$ diffeomorphism between open intervals, which may be unbounded. The change of variables formula holds for every nonnegative Lebesgue measurable function $f:J\to [0,\infty ]$, allowing infinite integrals. Applying it to $|f|$ also proves equivalence of Lebesgue integrability of $f$ and of the transformed integrand; for an integrable signed or complex $f$ the same identity then holds by splitting into positive and negative, or real and imaginary, parts. The absolute value of the derivative is essential for a decreasing substitution such as $\varphi (x)=1/x$. Endpoints can be omitted because they are null sets.