Change of Variable for Multi-dimensional Integral

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.