= Lebesgue substitution on an interval
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{title2=$\int_J f(y)\,dy=\int_I f(\phi(x))|\phi'(x)|\,dx$}
Let $\phi:I\to J$ 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 $\phi(x)=1/x$. Endpoints can be omitted because they are <null sets>.
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