= Monotone substitution inequality
{title2=$\int h(z(u))z^{\prime}(u)\,du\leq\int h$}
For a nondecreasing real function $z$ on an interval $I$ and nonnegative <measurable function> $h$,
$$
\int_I h(z(u))z^{\prime}(u)\,du\leq\int_{\mathbb R}h(s)\,ds.
$$
The pushforward of $z^{\prime}(u)\,du$ is dominated by <Lebesgue measure>: on each interval, the integral of the derivative is at most the increase of $z$. This handles jumps and singular parts without a smoothness assumption.
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