Change of Variables

This chapter gives a measurability property of the image of a measurable set under a Lipschitz mapping. It performs the change of variable in the integral of a uniformly continuous function, which is a step in the proof of the change of variable for an integrable measure. The chapter defines the image of a measure under a continuous bijection with a continuously differentiable inverse function. It expresses that a continuously differentiable change of variables, along with its inverse function, is an isomorphism of spaces of measures. The chapter shows that the image of an integral measure under a change of variable T is integrable. It also shows that the change of variable is an isomorphism of the spaces of integrable measures if it is continuously differentiable with bounded derivatives and if the same holds for its inverse.