  The change of variable theorem is proved under the sole hypothesis of
differentiability of the transformation. Specifically, it is shown under this
hypothesis that the transformed integral equals the given one over every
measurable subset on which the transformation is injective; that countably many
of these subsets cover the domain of invertibility; and that its complement --
the domain of non-invertibility -- is measurable and so may be broken off and
handled separately.

Classical Analysis and ODEs